![Page 1: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/1.jpg)
Stochastic Processes - lesson 3
Bo Friis Nielsen
Institute of Mathematical Modelling
Technical University of Denmark
2800 Kgs. Lyngby – Denmark
Email: [email protected]
![Page 2: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/2.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
![Page 3: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/3.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
![Page 4: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/4.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
? Lack of memory property
![Page 5: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/5.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
? Lack of memory property
• Continuous random variables
![Page 6: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/6.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
? Lack of memory property
• Continuous random variables
� Exponential distribution
![Page 7: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/7.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
? Lack of memory property
• Continuous random variables
� Exponential distribution
� Moments for continuous random variables
![Page 8: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/8.jpg)
Bo Friis Nielsen – 12/9-2000 2C04141
OutlineOutline
• Discrete random variables (from last lesson)
� Geometric distribution
? Lack of memory property
• Continuous random variables
� Exponential distribution
� Moments for continuous random variables
• Reading recommendations
![Page 9: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/9.jpg)
Bo Friis Nielsen – 12/9-2000 3C04141
• Bernoulli distribution
� Two possibilities succes(X = 1)/failure(X = 0)
• Binomial distribution
� Number of succeses in a sequence of independent
Bernoulli trials
� pdf (Probability Density Function)
? f(x) = P{X = x} =
n
x
px(1− p)n−x
� Mean and variance
? E(X) = np V (X) = np(1− p)
![Page 10: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/10.jpg)
Bo Friis Nielsen – 12/9-2000 4C04141
Poisson distributionPoisson distribution
• Number of faults/accidents/occurrences
• X ∈ P(λ) ⇔ P{X = x} = f(x) = λx
x!e−λ
• E(X) = V (X) = λ
![Page 12: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/12.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
![Page 13: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/13.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
![Page 14: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/14.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
![Page 15: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/15.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
• Possible values
![Page 16: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/16.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
• Possible values
� If we count all trials
![Page 17: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/17.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
• Possible values
� If we count all trials 1, . . . ,∞
![Page 18: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/18.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
• Possible values
� If we count all trials 1, . . . ,∞
� If we only count the failures
![Page 19: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/19.jpg)
Bo Friis Nielsen – 12/9-2000 5C04141
Geometric distributionGeometric distribution
• How many items produced to get one passing the quality
control
• Number of days to get rain
• Sequence of failures until the first success - sequence of
Bernoulli trials
• Possible values
� If we count all trials 1, . . . ,∞
� If we only count the failures 0, . . . ,∞
![Page 20: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/20.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
![Page 21: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/21.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS
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Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
![Page 23: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/23.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
![Page 24: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/24.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
• The probability of having x− 1 failures before the first
succes is
![Page 25: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/25.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
• The probability of having x− 1 failures before the first
succes is
P{x trials} =
![Page 26: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/26.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
• The probability of having x− 1 failures before the first
succes is
P{x trials} = P{x− 1 failures and then a succes} =
![Page 27: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/27.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
• The probability of having x− 1 failures before the first
succes is
P{x trials} = P{x− 1 failures and then a succes} =
(1− p)x−1p
![Page 28: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/28.jpg)
Bo Friis Nielsen – 12/9-2000 6C04141
Derivation of probability density function -
counting all trials
Derivation of probability density function -
counting all trials
• Let’s look at the sequence FFFFS with probability
(1− p)4p
• A general sequence will be like FFF. . .FFS
• The probability of having x− 1 failures before the first
succes is
P{x trials} = P{x− 1 failures and then a succes} =
(1− p)x−1p
• The cumulative distribution can be found to be
� F (x) =∑x
t=1(1− p)t−1p = 1− (1− p)x
![Page 30: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/30.jpg)
Bo Friis Nielsen – 12/9-2000 7C04141
• E(X) = 1p
V (X) = 1−p
p2
The expressions when counting only failuresThe expressions when counting only failures
• f(x) = (1− p)xp
• F (x) = 1− (1− p)x+1
• E(X) = 1−p
p
• V (X) = 1−p
p2
![Page 31: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/31.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
![Page 32: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/32.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
![Page 33: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/33.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
![Page 34: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/34.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
![Page 35: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/35.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n}
![Page 36: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/36.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}
![Page 37: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/37.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
![Page 38: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/38.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}
![Page 39: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/39.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
![Page 40: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/40.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
![Page 41: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/41.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
= (1−p)x+n
(1−p)n
![Page 42: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/42.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
= (1−p)x+n
(1−p)n= (1− p)x
![Page 43: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/43.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
= (1−p)x+n
(1−p)n= (1− p)x = P{X > x}
![Page 44: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/44.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
= (1−p)x+n
(1−p)n= (1− p)x = P{X > x}
• That is, the probability of exceeding x + n having reached
n is the same as the property of exceeding x starting from
the beginning.
![Page 45: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/45.jpg)
Bo Friis Nielsen – 12/9-2000 8C04141
The memoryless propertyThe memoryless property
• What will happen to the distribution knowing that n
failures already occured?
• That is we have been waiting for an empty cab and have
experienced 7 occupied
• Formally
P{X > x + n|X > n} = P{X>x+n∩X>n}P{X>n}
= P{X>x+n}P{X>n}
=1−(1−(1−p)x+n)1−(1−(1−p)n)
= (1−p)x+n
(1−p)n= (1− p)x = P{X > x}
• That is, the probability of exceeding x + n having reached
n is the same as the property of exceeding x starting from
the beginning. In other words no aging
![Page 46: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/46.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
![Page 47: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/47.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
![Page 48: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/48.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
![Page 49: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/49.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
• The probability density function in the continuous case
![Page 50: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/50.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
• The probability density function in the continuous case
� f(x) = F ′(x)
![Page 51: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/51.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
• The probability density function in the continuous case
� f(x) = F ′(x)
� Probabilistic interpretation
![Page 52: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/52.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
• The probability density function in the continuous case
� f(x) = F ′(x)
� Probabilistic interpretation
P{x ≤ X ≤ x + dx} = f(x)dx
![Page 53: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/53.jpg)
Bo Friis Nielsen – 12/9-2000 9C04141
Continuous random variablesContinuous random variables
• Cumulative distribution function, once again
� P{X ≤ x} = F (x)
• The probability density function in the continuous case
� f(x) = F ′(x)
� Probabilistic interpretation
P{x ≤ X ≤ x + dx} = f(x)dx or F (x) =∫ x−∞ f(t)dt
![Page 54: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/54.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
![Page 55: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/55.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
• F (x) = 1− e−λx
![Page 56: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/56.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
• F (x) = 1− e−λx ⇔ X ∈ exp(λ)
![Page 57: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/57.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
• F (x) = 1− e−λx ⇔ X ∈ exp(λ)
• f(x) = λe−λx
![Page 58: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/58.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
• F (x) = 1− e−λx ⇔ X ∈ exp(λ)
• f(x) = λe−λx
• The exponential is without memory like the geometric
distribution
![Page 59: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/59.jpg)
Bo Friis Nielsen – 12/9-2000 10C04141
The continuous parallel to the geometric
distribution
the exponential distribution
The continuous parallel to the geometric
distribution
the exponential distribution
• F (x) = 1− e−λx ⇔ X ∈ exp(λ)
• f(x) = λe−λx
• The exponential is without memory like the geometric
distribution
• The geometric/exponential distributions are the unique
memoryless distributions
![Page 60: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/60.jpg)
Bo Friis Nielsen – 12/9-2000 11C04141
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Exponential density with mean=1
’exppdf.lst’
![Page 61: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/61.jpg)
Bo Friis Nielsen – 12/9-2000 11C04141
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Exponential density with mean=1
’exppdf.lst’
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Exponential cumulative distribution with mean=1
’expcdf.lst’
![Page 62: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/62.jpg)
Bo Friis Nielsen – 12/9-2000 12C04141
The normal densityThe normal density
0
0.05
0.1
0.15
0.2
0.25
0.3
-4 -3 -2 -1 0 1 2 3 4
Normal density with mean=0 and variance=1
’normpdf.lst’
![Page 64: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/64.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
![Page 65: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/65.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx
![Page 66: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/66.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
![Page 67: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/67.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx
![Page 68: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/68.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx = 1
λby partial integration
![Page 69: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/69.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx = 1
λby partial integration
• The variance of continuous random variables
![Page 70: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/70.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx = 1
λby partial integration
• The variance of continuous random variables
� V (X) =∫∞−∞(x− µ)2f(x)dx
![Page 71: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/71.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx = 1
λby partial integration
• The variance of continuous random variables
� V (X) =∫∞−∞(x− µ)2f(x)dx = σ2
![Page 72: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/72.jpg)
Bo Friis Nielsen – 12/9-2000 13C04141
Moments revisitedMoments revisited
• The mean for continuous random variables
� E(X) =∫∞−∞ xf(x)dx = µ
�∫∞0 xλe−λxdx = 1
λby partial integration
• The variance of continuous random variables
� V (X) =∫∞−∞(x− µ)2f(x)dx = σ2
� X ∈ exp(λ) V (X) = 1λ2
![Page 73: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/73.jpg)
Bo Friis Nielsen – 12/9-2000 14C04141
Small examples using the exponential
distribution
Small examples using the exponential
distributionThe time between two consecutive attempts access attempts
to a popular web site can be adequately described by an
exponential distribution with mean 5 seconds.
![Page 74: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/74.jpg)
Bo Friis Nielsen – 12/9-2000 14C04141
Small examples using the exponential
distribution
Small examples using the exponential
distributionThe time between two consecutive attempts access attempts
to a popular web site can be adequately described by an
exponential distribution with mean 5 seconds.
• What is the variance of the time between two consecutive
access attempts?
![Page 75: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/75.jpg)
Bo Friis Nielsen – 12/9-2000 14C04141
Small examples using the exponential
distribution
Small examples using the exponential
distributionThe time between two consecutive attempts access attempts
to a popular web site can be adequately described by an
exponential distribution with mean 5 seconds.
• What is the variance of the time between two consecutive
access attempts?
• What is the probability that the time between two
consecutive attempts will be less than 1 second?
![Page 76: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/76.jpg)
Bo Friis Nielsen – 12/9-2000 14C04141
Small examples using the exponential
distribution
Small examples using the exponential
distributionThe time between two consecutive attempts access attempts
to a popular web site can be adequately described by an
exponential distribution with mean 5 seconds.
• What is the variance of the time between two consecutive
access attempts?
• What is the probability that the time between two
consecutive attempts will be less than 1 second?
• Knowing that no attempt has been made within the last
minute, what is the probability that there won’t be a new
attempt within the next second?
![Page 78: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/78.jpg)
Bo Friis Nielsen – 12/9-2000 15C04141
ModelModel
• X : the time between two consecutive attempts
![Page 79: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/79.jpg)
Bo Friis Nielsen – 12/9-2000 15C04141
ModelModel
• X : the time between two consecutive attempts
• X ∈ exp(λ)
![Page 80: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/80.jpg)
Bo Friis Nielsen – 12/9-2000 15C04141
ModelModel
• X : the time between two consecutive attempts
• X ∈ exp(λ)
• E(X) = 1λ
= 5s
![Page 81: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/81.jpg)
Bo Friis Nielsen – 12/9-2000 15C04141
ModelModel
• X : the time between two consecutive attempts
• X ∈ exp(λ)
• E(X) = 1λ
= 5s ⇒ λ = 0.2s−1
![Page 82: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/82.jpg)
Bo Friis Nielsen – 12/9-2000 15C04141
ModelModel
• X : the time between two consecutive attempts
• X ∈ exp(λ)
• E(X) = 1λ
= 5s ⇒ λ = 0.2s−1
• X ∈ exp(0.2)
![Page 83: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/83.jpg)
Bo Friis Nielsen – 12/9-2000 16C04141
What is the variance of the time between
two consecutive access attempts?
What is the variance of the time between
two consecutive access attempts?
![Page 84: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/84.jpg)
Bo Friis Nielsen – 12/9-2000 16C04141
What is the variance of the time between
two consecutive access attempts?
What is the variance of the time between
two consecutive access attempts?
• V (X) = 1λ2
![Page 85: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/85.jpg)
Bo Friis Nielsen – 12/9-2000 16C04141
What is the variance of the time between
two consecutive access attempts?
What is the variance of the time between
two consecutive access attempts?
• V (X) = 1λ2 = 0.04
![Page 86: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/86.jpg)
To exponential distribution
Bo Friis Nielsen – 12/9-2000 17C04141
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
![Page 87: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/87.jpg)
To exponential distribution
Bo Friis Nielsen – 12/9-2000 17C04141
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
• P{X ≤ 1}
![Page 88: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/88.jpg)
To exponential distribution
Bo Friis Nielsen – 12/9-2000 17C04141
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
• P{X ≤ 1} = F (1) = 1− exp−0.2·1
![Page 89: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/89.jpg)
To exponential distribution
Bo Friis Nielsen – 12/9-2000 17C04141
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
What is the probability that the time
between two consecutive attempts will be
less than 1 second?
• P{X ≤ 1} = F (1) = 1− exp−0.2·1 = 0.1813
![Page 90: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/90.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
![Page 91: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/91.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60}
![Page 92: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/92.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
![Page 93: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/93.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
= P{X>61}P{X>60}
![Page 94: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/94.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
= P{X>61}P{X>60}
=1−(1−exp−0.02·61)1−(1−exp−0.02·60)
![Page 95: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/95.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
= P{X>61}P{X>60}
=1−(1−exp−0.02·61)1−(1−exp−0.02·60)
= exp−0.02·61
exp−0.02·60
![Page 96: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/96.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
= P{X>61}P{X>60}
=1−(1−exp−0.02·61)1−(1−exp−0.02·60)
= exp−0.02·61
exp−0.02·60 = exp−0.02
![Page 97: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/97.jpg)
Bo Friis Nielsen – 12/9-2000 18C04141
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
Knowing that no attempt has been made
within the last minute, what is the
probability that there won’t be a new
attempt within the next second?
• P{X > 61|X > 60} = P{X>61∩X>60}P{X>60}
= P{X>61}P{X>60}
=1−(1−exp−0.02·61)1−(1−exp−0.02·60)
= exp−0.02·61
exp−0.02·60 = exp−0.02 = 0.8187
• Which is exactly the lack of memory property
![Page 98: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/98.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions.
![Page 99: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/99.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
![Page 100: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/100.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
� E(Y ) = E(g(X)) =∫∞−∞ g(x)f(x)dx
![Page 101: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/101.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
� E(Y ) = E(g(X)) =∫∞−∞ g(x)f(x)dx
• One important example
![Page 102: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/102.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
� E(Y ) = E(g(X)) =∫∞−∞ g(x)f(x)dx
• One important example
• E(X2) =∫∞−∞ x2f(x)dx
![Page 103: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/103.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
� E(Y ) = E(g(X)) =∫∞−∞ g(x)f(x)dx
• One important example
• E(X2) =∫∞−∞ x2f(x)dx = V (X) + (E(X))2
![Page 104: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/104.jpg)
Bo Friis Nielsen – 12/9-2000 19C04141
A unified and more general look at momentsA unified and more general look at moments
• Sometimes convenient to calculate similar
integral/expectation for other functions. e.g. the mean of
the random variable Y = g(X)
� E(Y ) = E(g(X)) =∫∞−∞ g(x)f(x)dx
• One important example
• E(X2) =∫∞−∞ x2f(x)dx = V (X) + (E(X))2 = σ2 + µ2
![Page 105: Stochastic Processes - lesson 3 · Stochastic Processes - lesson 3 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby { Denmark](https://reader034.vdocuments.net/reader034/viewer/2022051815/603fc0384a600065cb23c411/html5/thumbnails/105.jpg)
Bo Friis Nielsen – 12/9-2000 20C04141
Reading recommendationsReading recommendations
• for Friday September 8 and Tuesday September 12, read
Chapter 2 lightly, Chapter 3 section: 3.1,3.2,3.3,3.5,
Chapter 4 section: 4.1,4.3,4.4
• For Friday September 15, read 3.8-3.10.
• News on exercises for Friday, not later than Thursday 3pm
on the net.