iam 530 elements of probability and …osezgin/iam 530-lecture4.pdfbinomial theorem •for any real...
TRANSCRIPT
![Page 1: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/1.jpg)
IAM 530 ELEMENTS OF PROBABILITY AND
STATISTICS
LECTURE 4-SOME DISCERETE AND CONTINUOUS DISTRIBUTION
FUNCTIONS
![Page 2: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/2.jpg)
SOME DISCRETE PROBABILITY DISTRIBUTIONS
Degenerate, Uniform, Bernoulli, Binomial, Poisson, Negative Binomial,
Geometric, Hypergeometric
![Page 3: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/3.jpg)
DEGENERATE DISTRIBUTION
• An rv X is degenerate at point k if
1,
0, . .
X kP X x
o w
The cdf: 0,
1,
X kF x P X x
X k
![Page 4: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/4.jpg)
UNIFORM DISTRIBUTION
• A finite number of equally spaced values are equally likely to be observed.
• Example: throw a fair die. P(X=1)=…=P(X=6)=1/6
,...2,1N;N,...,2,1x;N
1)xX(P
12
)1N)(1N()X(Var;
2
1N)X(E
![Page 5: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/5.jpg)
BERNOULLI DISTRIBUTION • An experiment consists of one trial. It can result in one of 2
outcomes: Success or Failure (or a characteristic being Present or Absent).
• Probability of Success is p (0<p<1)
)1(
)1()()(
1)1(0
1)1(0)()(
222
222
1
0
pp
ppppXEXEXV
pppXE
pppyxpXEy
1 with probability ;0 1
0 with probability 1
pX p
p
1p0and;1,0xfor)p1(p)xX(P x1x
![Page 6: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/6.jpg)
Binomial Experiment • Experiment consists of a series of n identical trials
• Each trial can end in one of 2 outcomes: Success or Failure
• Trials are independent (outcome of one has no bearing on outcomes of others)
• Probability of Success, p, is constant for all trials
• Random Variable X, is the number of Successes in the n trials is said to follow Binomial Distribution with parameters n and p
• X can take on the values x=0,1,…,n
• Notation: X~Bin(n,p)
![Page 7: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/7.jpg)
3
2
2
3
)1()0()0()(0
)1(3)1()1()(1,,
)1(3)2()2()(2,,
)3()3()(3
:Trials 3 with experimentan of outcomesConsider
ppYPFFFPyFFF
pppYPFFSFSFSFFPyFFSFSFSFF
pppYPFSSSFSSSFPyFSSSFSSSF
ppYPSSSPySSS
nxppx
nxpxXP
ppFxnSx
xnx
n
x
nnFxnSx
xnx
xnxss
ss
,...,1,0)1()()()3
)1( ) )( (and oft arrangemeneach ofy Probabilit )2
)!(!
! positions of sequence ain ) )( (and arranging of waysof # 1)
:GeneralIn
![Page 8: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/8.jpg)
• Example:
• There are black and white balls in a box. Select and record the color of the ball. Put it back and re-pick (sampling with replacement).
• n: number of independent and identical trials
• p: probability of success (e.g. probability of picking a black ball)
• X: number of successes in n trials
![Page 9: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/9.jpg)
BINOMIAL THEOREM
• For any real numbers x and y and integer n>0
inin
i
n yxi
nyx
0
)(
![Page 10: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/10.jpg)
• If X~Bin(n,p), then
p)-np(1Var(X)
npE(X)
![Page 11: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/11.jpg)
POISSON DISTRIBUTION
• The number of occurrences in a given time interval can be modeled by the Poisson distribution.
• e.g. waiting for bus, waiting for customers to arrive in a bank.
• Another application is in spatial distributions.
• e.g. modeling the distribution of bomb hits in an area or the distribution of fish in a lake.
![Page 12: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/12.jpg)
POISSON DISTRIBUTION
• If X~ Poisson(λ), then
• E(X)= Var(X)=λ
,...2,1,0!
)( xx
exp
x
![Page 13: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/13.jpg)
RELATIONSHIP BETWEEN BINOMIAL AND POISSON
~ , with mgf 1nt
XX Bin n p M t pe p
Let =np.
1
lim lim 1
1lim 1
nt
Xn n
ntte
Yn
M t pe p
ee M t
n
The mgf of Poisson( )
The limiting distribution of Binomial rv is the
Poisson distribution.
![Page 14: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/14.jpg)
NEGATIVE BINOMIAL DISTRIBUTION (PASCAL OR WAITING TIME DISTRIBUTION)
• X: number of Bernoulli trials required to get a fixed number of failures before the r th success; or, alternatively,
• Y: number of Bernoulli trials required to get a fixed number of successes, such as r successes.
![Page 15: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/15.jpg)
NEGATIVE BINOMIAL DISTRIBUTION (PASCAL OR WAITING TIME DISTRIBUTION)
X~NB(r,p)
1p0,...;1,0x;)p1(px
1xr)xX(P xr
2p
)p1(r)X(Var
p
)p1(r)X(E
![Page 16: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/16.jpg)
NEGATIVE BINOMIAL DISTRIBUTION
• An alternative form of the pdf:
Note: Y=X+r
1p0,...;1r,ry;)p1(p1r
1y)yY(P ryr
2p
)p1(r)X(Var)Y(Var
p
rr)X(E)Y(E
![Page 17: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/17.jpg)
GEOMETRIC DISTRIBUTION
• Distribution of the number of Bernoulli trials required to get the first success.
• Used to model the number of Bernoulli trials needed until the first Success occurs (P(S)=p)
– First Success on Trial 1 S, y = 1 p(1)=p
– First Success on Trial 2 FS, y = 2 p(2)=(1-p)p
– First Success on Trial k F…FS, y = k p(k)=(1-p)k-1 p
![Page 18: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/18.jpg)
• It is the special case of the Negative Binomial Distribution r=1.
1
1 , 1,2,x
P X x p p x
X~Geometric(p)
2p
)p1()X(Var
p
1)X(E
![Page 19: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/19.jpg)
• Example: If probability is 0.001 that a light bulb will fail on any given day, then what is the probability that it will last at least 30 days?
• Solution:
97.0)999.0()001.01(001.0)30X(P 30
31x
1x
![Page 20: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/20.jpg)
HYPERGEOMETRIC DISTRIBUTION
• A box contains N marbles. Of these, M are red. Suppose that n marbles are drawn randomly from the box without replacement. The distribution of the number of red marbles, x is
, 0,1,...,
M N M
x n xP X x x n
N
n
It is dealing with finite population.
X~Hypergeometric(N,M,n)
![Page 21: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/21.jpg)
Example 1 As voters exit the polls, you ask a representative random sample of 6 voters if they voted for proposition 100. If the true percentage of voters who vote for the proposition is 55.1%, what is the probability that, in your sample, exactly 2 voted for the proposition and 4 did not?
PRACTICE PROBLEMS
6
2
P(2 yes votes exactly) = (.551)2 (.449)4 = 18.5%
![Page 22: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/22.jpg)
Example 2 You are performing a cohort study. If the probability of developing disease in the exposed group is .05 for the study duration, then if you sample (randomly) 500 exposed people, how many do you expect to develop the disease? Give a margin of error (+/- 1 standard deviation) for your estimate.
X ~ binomial (500, .05) E(X) = 500 (.05) = 25 Var(X) = 500 (.05) (.95) = 23.75 StdDev(X) = square root (23.75) = 4.87 25 4.87
![Page 23: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/23.jpg)
Example 3 Patients arrive at the emergency room of Hospital A at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?
= 6/hour = 3/half-hour, x = 4
4 33 (2.71828)(4) .1680
4!f
![Page 24: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/24.jpg)
Example 4 Ahmet has removed two dead batteries from his camera and put them into his drawer. In the drawer there are also two good batteries. The four batteries look identical. Ahmet need battery and now randomly selects two of the four batteries. What is the probability he selects the two good batteries?
2 2 2! 2!
2 0 2!0! 0!2! 1( ) .167
4 4! 6
2 2!2!
f x
![Page 25: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/25.jpg)
Example 5 At “busy time” a telephone exchange is very near capacity, so callers have difficulty placing their calls. It may be on interest to know the number of attempts necessary in order to gain a connection. Suppose that we let p = 0.05 be the probability of a connection during a busy time. We are interested in knowing the probability that 5 attempts are necessary for a successful call.
![Page 26: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/26.jpg)
The random variable X is the number of attempts for
a successful call. Then
X~geometric(0.05),
• So that for with x = 5 and p = 0.05 yields:
• P(X=x) = geometric(5;0.05)
• = (0.05)(0.95)4
• = 0.041
• And the expected number of attempts is 1
200.05
![Page 27: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/27.jpg)
Example 6 Suppose that a company wishes to hire
three new workers and each applicant interviewed
has a probability of 0.6 of being found acceptable.
What is the probability that exactly six applicants
need to be interviewed?
The distribution of the total number of applicants
that the company needs to interview Negative
Binomial distribution with parameter p = 0.6 and r =
3.
3 35
6 0.4 0.6 0.1382
P X
![Page 28: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/28.jpg)
SOME CONTINUOUS PROBABILITY DISTRIBUTIONS
Uniform, Normal, Exponential, Gamma, Chi-Square, Beta
Distributions
![Page 29: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/29.jpg)
A random variable X is said to be uniformly distributed if its density function is
2
1( ) .
( )E(X) (X)
2 12
f x a x bb a
a b b aV
UNIFORM DISTRIBUTION
0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
f x
x0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
f x
x
1
b a
a b
f x
x
![Page 30: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/30.jpg)
Example
– The daily sale of gasoline is uniformly distributed between 2,000 and 5,000 gallons. Find the probability that sales are:
– Between 2,500 and 3,000 gallons
– More than 4,000 gallons
– Exactly 2,500 gallons
f(x) = 1/(5000-2000) = 1/3000 for x: [2000,5000]
P(2500 X 3000) = (3000-2500)(1/3000) = .1667
P(X 4000) = (5000-4000)(1/3000) = .333
P(X=2500) = (2500-2500)(1/3000) = 0
![Page 31: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/31.jpg)
NORMAL DISTRIBUTION
• This is the most popular continuous distribution.
– Many distributions can be approximated by a normal distribution.
– The normal distribution is the cornerstone distribution of statistical inference.
![Page 32: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/32.jpg)
• A random variable X with mean and variance is normally distributed if its probability density function is given by
2
(1/2)1( ) ; 0
2
3.14159... 2.71828...
x
f x e x
where and e
![Page 33: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/33.jpg)
THE SHAPE OF THE NORMAL DISTRIBUTION
The normal distribution is bell shaped, and
symmetrical around
![Page 34: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/34.jpg)
• Two facts help calculate normal probabilities: – The normal distribution is symmetrical.
– Any normal distribution can be transformed into a specific normal distribution called…
“STANDARD NORMAL DISTRIBUTION”
FINDING NORMAL PROBABILITIES
![Page 35: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/35.jpg)
STANDARD NORMAL DISTRIBUTION
• NORMAL DISTRIBUTION WITH MEAN 0 AND VARIANCE 1.
• IF X~N( , 2), THEN
~ (0,1)
XZ N
![Page 36: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/36.jpg)
• Example
The amount of time it takes to assemble a computer is normally distributed, with a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability that a computer is assembled in a time between 45 and 60 minutes?
• Solution – If X denotes the assembly time of a computer, we seek
the probability P(45<X<60).
– This probability can be calculated by creating a new normal variable the standard normal variable.
![Page 37: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/37.jpg)
P(45<X<60) = P( < < ) 45 X 60 - 50 - 50
10 10
= P(-0.5 < Z < 1)
To complete the calculation we need to compute the probability under the standard normal distribution
~ (0,1)X
Z N
By using following transformation.
![Page 38: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/38.jpg)
STANDARD NORMAL TABLE 1
![Page 39: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/39.jpg)
STANDARD NORMAL TABLE 2
![Page 40: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/40.jpg)
STANDARD NORMAL TABLE 3
![Page 41: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/41.jpg)
For P(-.5 < Z < 1) We need to find the shaded area
z0 = 1 z0 = -.5
= P(-.5<Z<1) = P(-.5<Z<0)+ P(0<Z<1)
P(-.5<Z<1) = P(-.5<Z<0)+ P(0<Z<1) = .1915 + .3413 = .5328
![Page 42: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/42.jpg)
10% 0%
P(X< 0 ) = P(Z< ) = P(Z< - 2) 0 - 10
5
=P(Z>2) =
X
Example
– The rate of return (X) on an investment is normally distributed with a mean of 10% and standard deviation of 5%
– What is the probability of losing money?
0.5 - P(0<Z<2) = 0.5 - .4772 = .0228
![Page 43: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/43.jpg)
STANDARDIZATION FORMULA • If X~N( , 2), then the standardized value Z of
any ‘X-score’ associated with calculating probabilities for the X distribution is:
• The standardized value Z of any ‘X-score’ associated with calculating probabilities for the X distribution is:
• (Converse Formula)
XZ
.x z
![Page 44: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/44.jpg)
• Sometimes we need to find the value of Z for a given probability
• We use the notation zA to express a Z value for which P(Z > zA) = A
FINDING VALUES OF Z
zA
A
![Page 45: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/45.jpg)
PERCENTILE • The pth percentile of a set of measurements is
the value for which at most p% of the measurements are less than that value.
• 80th percentile means P( Z < a ) = 0.80
• If Z ~ N(0,1) and A is any probability, then
P( Z > zA) = A
A
zA
![Page 46: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/46.jpg)
• Example – Determine z exceeded by 5% of the population
– Determine z such that 5% of the population is below
• Solution z.05 is defined as the z value for which the area on its right
under the standard normal curve is .05.
0.05
Z0.05 0
0.45
1.645
0.05
-Z0.05
![Page 47: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/47.jpg)
EXAMPLES
• Let X be rate of return on a proposed investment. Mean is 0.30 and standard deviation is 0.1.
a) P(X>.55)=?
b) P(X<.22)=?
c) P(.25<X<.35)=?
d) 80th Percentile of X is?
e) 30th Percentile of X is?
Standardization formula
Converse Formula
![Page 48: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/48.jpg)
ANSWERS a) b) c) d) e)
80th Percentile of X is
30th Percentile of X is
![Page 49: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/49.jpg)
THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION
• The normal distribution provides a close approximation to the Binomial distribution when n (number of trials) is large and p (success probability) is close to 0.5.
• The approximation is used only when
np 5 and
n(1-p) 5
![Page 50: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/50.jpg)
• If the assumptions are satisfied, the Binomial random variable X can be approximated by normal distribution with mean = np and 2 = np(1-p).
• In probability calculations, the continuity correction improves the results. For example, if X is Binomial random variable, then
P(X a) ≈ P(X<a+0.5)
P(X a) ≈ P(X>a-0.5)
![Page 51: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/51.jpg)
EXAMPLE
• Let X ~ Binomial(25,0.6), and want to find P(X ≤ 13).
• Exact binomial calculation:
• Normal approximation (w/o correction): Y~N(15,2.45²)
13
0x
x25x 267.0)4.0()6.0(x
25)13X(P
206.0)82.0Z(P)45.2
1513Z(P)13Y(P)13X(P
Normal approximation is good, but not great!
![Page 52: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/52.jpg)
EXPONENTIAL DISTRIBUTION
• The exponential distribution can be used to model – the length of time between telephone calls – the length of time between arrivals at a service
station – the lifetime of electronic components.
• When the number of occurrences of an event follows the Poisson distribution, the time between occurrences follows the exponential distribution.
![Page 53: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/53.jpg)
A random variable is exponentially distributed if its probability density function is given by
f(x) = e- x, x>=0.
is the distribution parameter >0).
E(X) = V(X) = 2
The cumulative distribution function is F(x) =1 e-x/ , x 0
/1, 0, 0x
Xf x e x
is a distribution parameter.
![Page 54: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/54.jpg)
0
0.5
1
1.5
2
2.5
a b
P(a<X<b) = e-a/ e-b/
• Finding exponential probabilities is relatively easy:
– P(X < a) = P(X ≤ a)=F(a)=1 – e –a/
– P(X > a) = e–a/
– P(a< X < b) = e – a/ – e – b/
![Page 55: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/55.jpg)
Example The service rate at a supermarket checkout is 6 customers
per hour. – If the service time is exponential, find the following
probabilities: • A service is completed in 5 minutes, • A customer leaves the counter more than 10
minutes after arriving • A service is completed between 5 and 8 minutes.
• Solution
– A service rate of 6 per hour A service rate of .1 per minute ( = .1/minute).
– P(X < 5) = 1-e-.lx = 1 – e-.1(5) = .3935
– P(X >10) = e-.lx = e-.1(10) = .3679
– P(5 < X < 8) = e-.1(5) – e-.1(8) = .1572
![Page 56: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/56.jpg)
• X~ Gamma( , )
GAMMA DISTRIBUTION
1 /1, 0, 0, 0xf x x e x
2 and E X Var X
11 , M t t t
![Page 57: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/57.jpg)
• Gamma Function:
1
0
xx e dx
where is a positive integer.
Properties:
1 , 0
1 ! for any integer 1n n n
1
2
![Page 58: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/58.jpg)
• Let X1,X2,…,Xn be independent rvs with Xi~Gamma( i, ). Then,
1 1
~ ,n n
i ii i
X Gamma
•Let X be an rv with X~Gamma( , ). Then, ~ , where is positive constant.cX Gamma c c
• Let X1,X2,…,Xn be a random sample with Xi~Gamma( , ). Then,
1 ~ ,
n
ii
XX Gamma n
n n
![Page 59: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/59.jpg)
• Special cases: Suppose X~Gamma(α,β)
– If α=1, then X~ Exponential(β)
– If α=p/2, β=2, then X~ 2 (p) (will come back in a min.)
– If Y=1/X, then Y ~ inverted gamma.
![Page 60: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/60.jpg)
CHI-SQUARE DISTRIBUTION
• X~ 2( )= Gamma( /2,2)
and 2E X Var X
Chi-square with degrees of freedom
2/1t)t21()t(M 2/p
,...2,1,0,)2/(2
1)( 2/12/
2/xexxf x
![Page 61: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/61.jpg)
DEGREES OF FREEDOM
• In statistics, the phrase degrees of freedom is used to describe the number of values in the final calculation of a statistic that are free to vary.
• The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df) .
• How many components need to be known before the vector is fully determined?
![Page 62: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/62.jpg)
• If rv X has Gamma( , ) distribution, then Y=2X/ has Gamma( ,2) distribution. If 2 is positive integer, then Y has distribution. 2
2
•Let X1,X2,…,Xn be a r.s. with Xi~N(0,1). Then,
2 2
1
~n
i ni
X
•Let X be an rv with X~N(0, 1). Then,
2 2
1~X
![Page 63: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/63.jpg)
BETA DISTRIBUTION
• The Beta family of distributions is a continuous family on (0,1) and often used to model proportions.
1111 , 0 1, 0, 0.
,f x x x x
B
where 1
1
0
, 1B x x dx
2 and 1
E X Var X
![Page 64: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/64.jpg)
CAUCHY DISTRIBUTION • It is a symmetric and bell-shaped distribution
on ( , ) with pdf
E XSince , the mean does not exist.
• The mgf does not exist.
• measures the center of the distribution and it is the median.
• If X and Y have N(0,1) distribution, then Z=X/Y has a Cauchy distribution with =0 and σ=1.
0,
)x
(1
11)x(f
2
![Page 65: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/65.jpg)
LOG-NORMAL DISTRIBUTION
• An rv X is said to have the lognormal distribution, with parameters µ and 2, if Y=ln(X) has the N(µ, 2) distribution.
•The lognormal distribution is used to model continuous random quantities when the distribution is believed to be skewed, such as certain income and lifetime variables.
![Page 66: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/66.jpg)
STUDENT’S T DISTRIBUTION
• This distribution will arise in the study of population mean when the underlying distribution is normal.
• Let Z be a standard normal rv and let U be a chi-square distributed rv independent of Z, with degrees of freedom. Then,
~/
ZX t
U
When n , X N(0,1).
![Page 67: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/67.jpg)
F DISTRIBUTION
• Let U and V be independent rvs with chi-square distributions with 1 and 2 degrees of freedom. Then,
1 2
1
,
2
/~
/
UX F
V
![Page 68: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/68.jpg)
MOMENT GENERATING FUNCTION
xall
tx
xall
tx
tXX
discreteisXif)x(fe
.contisXifdx)x(fe
)e(E)t(M
The moment generating function (m.g.f.) of random variable X is defined as
for t Є (-h,h) for some h>0.
![Page 69: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/69.jpg)
PROPERTIES OF M.G.F.
• M(0)=E[1]=1
• If a r.v. X has m.g.f. MX(t), then Y=aX+b has a m.g.f.
•
• M.g.f does not always exists (e.g. Cauchy distribution)
( )bt
Xe M at
( ) ( )( ) (0) .k k k th
X XE X M where M is the k derivative
![Page 70: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/70.jpg)
2 3
0
2 2
2
2
Consider the series expansion of :
1 ...! 2 6
Note that by taking derivatives with respect to , we get:
2 30 1 ... 1 ...
2! 3! 2!
20 1 ...
2!
Now, Replacing
x
ix
i
xx
x
e
x x xe x
i
x
de x x xx e
dx
d e x
dx
x
2 3
0
2 2 3 2 32
0 0 0
22
2
0
with , we get:
( ) ( ) ( )1 ...
! 2 6
Taking derivatives with respect to and evaluating at 0 :
2 30 ... ... 0 0 ...
2! 3! 2!
0
itX
i
tX
t t t
tX
t
tX
tX tX tXe tX
i
t t
de tX t X t XX X tX X X
dt
d eX tX
dt
3 2 2
0
2 ( )
0 0 0
... 0 ...
'( ) (X), ''( ) , ... ( )
t
k K
X X Xt t t
X X
M t E M t E X M t E X
![Page 71: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/71.jpg)
BINOMIAL DISTRIBUTION
0
0
1 1
2 1
( ) (1 )
(1 ) (1 )
'( ) (1 ) (1 )
''( ) ( 1) (1 ) (1 )
(X) '(0) (1) (1 )
ntX tx x n x
y
nx n
t n x t
y
n nt t t t
n nt t t t t
n
nM t E e e p p
x
npe p pe p
x
M t n pe p pe np pe p e
M t np n pe p pe e pe p e
E M np p p1
2 12
2 2 2 2 2
22 2 2 2
(1)
''(0) ( 1) (1) (1 ) (1) (1) (1) (1 ) [1]
( 1) 1 (1 )
(X) (X) (1 ) ( ) (1 )
n n
np
E X M np n p p p p p
np n p n p np np n p np p
V E X E n p np p np np p
![Page 72: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/72.jpg)
GEOMETRIC DISTRIBUTION
1
1 1 1
1
1
2 3
2
3 2
( )
1 1 (1 )
(1 (1 ) )'( ) ''( )
(1 (1 ) ) (1 (1 ) )
1 (1 (1 )) (1 (1 ))(X) '(0) (X ) ''(0)
(1 (1 ))
xtX tx x tx y t
y y y
t t tx
t
t ty
t t t
t t
p pM t E e e q p e q qe
q q
pqe pe peqe
q qe p e
pe pe p eM t M t
p e p e
p p pE M E M
p p p
2 2
2 2 2
(1 (1 )) 1 (1 )( ) ( ) E(X)
p pVar X E X
p p p
![Page 73: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/73.jpg)
POISSON DISTRIBUTION
0 0
1
0
1 1 1 2
2 2
2 2
( )x! x!
x!
'( ) ''( ) ( )
(X) E(X )
var( )
tt
t t t
xty
tX tx
y y
xt
ee
y
e e et t t
e eeM t E e e
ee e e e
M t e e M t e e e e
E
X
![Page 74: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/74.jpg)
EXPONENTIAL DISTRIBUTION 1
0 0
1
* *
0 0
* ** 1
* 0
2 2
1 1( )
1 1where
1
1 1 1( ) (0 1) (1 )
1 1
'( ) 1(1 ) ( ) (1 )
''( ) 2 (1
x ttX tx x
tx
x
x
M t E e e e dx e dx
e dx e dxt
M t e tt
M t t t
M t 3 2 3
2 2 2 2
) ( ) 2 (1 )
(X) '(0)
(X) ''(0) '(0) 2
t t
E M
V M M
![Page 75: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/75.jpg)
GAMMA DISTRIBUTION
1
0
1 1
1 1
0 0
1 * *
0
*
1
1( )
( )
1 1
( ) ( )
1 where
( ) 1
1( ) ( ) (1 )
( )
'( ) (1 ) ( )
tX tx x
tx t x
x
M t E e e x e dx
x e dx x e dx
x e dxt
M t t
M t t 1
2 2 2
2 2 2 2
(1 )
''( ) ( 1) (1 ) ( ) ( 1) (1 )
(X) '(0)
(X) ''(0) '(0) ( 1) ( )
t
M t t t
E M
V M M
![Page 76: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/76.jpg)
NORMAL DISTRIBUTION 2 2 2
2 2 2 22 2
2 2 2
2 2 22
22 2 2 2 2
1 1 1( ) exp exp
2 2 22 2
1 x( )exp
2 22
Completing the square: ( ) 2
1( )
2
tX txx x x
M t E e e dx tx dx
x tdx
t t t
M t
2 22 2 2 22 2 2
2 2 2 2 22
2 22 2 2 2 22 2
2 2 2 22
222 22
2 22
2 2x( )exp
2 2 2 2
2 21 x( )exp
2 2 22
2( )1 1exp
2 22
t t t tx tdx
t t t tx tdx
t tx t
2 22 2 2
2 22
2 2
2 ( )1 1exp exp
2 22
The last integral being 1, since it is integrating over the density of a normal R.V.: ~ ,
( ) ex
dx
t t x tdx
Y N t
M t
22 2 2 2
2
2p exp
2 2
t t tt
![Page 77: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/77.jpg)
CHARACTERISTIC FUNCTION
xall
itx
xall
itx
itX
X
discreteisXifxfe
contisXifdxxfe
eEt)(
.)(
)()(
The c.h.f. of random variable X is defined as
for all real numbers t.
C.h.f. always exists.
1,12 ii
![Page 78: IAM 530 ELEMENTS OF PROBABILITY AND …osezgin/IAM 530-LECTURE4.pdfBINOMIAL THEOREM •For any real numbers x and y and integer n>0 i n i n i nx y i n 0 ( ) •If X~Bin(n,p), then](https://reader031.vdocuments.net/reader031/viewer/2022022501/5aa8f1847f8b9a77188c2d0e/html5/thumbnails/78.jpg)
Examples
• Binomial Distribution:
• Poisson Distribution:
• Negative Binomial Dist.:
• Exponential Dist.:
• Gamma Dist.:
• Normal Dist.:
• Chi-sqr Dist.:
npitpetC )1)(()(
))1)((()( iteetC
kk itqeptC ))(1()(
1C(t) (1 )it
C(t) (1 ) rit
2 2
C( ) ( )2
tt e it
1
2( ) (1 2 )n
C t it