intro stats for engineering class stat 235 -...
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Intro Stats for engineeringClass STAT 235 - ES1
Department of Mathematical and Statistical Sciences, UofA
Matúš MaciakOffice Hours: T R 13:00 - 14:00 (or by appointment)@ Central Academy Building (CAB) 460
1 / 13STAT 235 | Lecture 4
N
Random Variable Characteristics
Random Variable Characteristicso Expected Value of a Random Variable
E(X) =
n∑i=1
xi · P[X = xi ] E(X) =
∫ ∞−∞
x · f (x)dx
o Variance of a Random Variable
Var(X) = E[X − E(X)]2 Var(X) = E[X − E(X)]2
↪→ where the second moment is obtained as follows:
E(X 2) =
n∑i=1
x2i · P[X = xi ] E(X 2) =
∫ ∞−∞
x2 · f (x)dx
o In addition, the following always holds:
Eg(X) =
n∑i=1
g(xi) · P[X = xi ] Eg(X) =
∫ ∞−∞
g(x) · f (x)dx
2 / 13STAT 235 | Lecture 4
N
Random Variable Characteristics
Random Variable Characteristicso Expected Value of a Random Variable
E(X) =
n∑i=1
xi · P[X = xi ] E(X) =
∫ ∞−∞
x · f (x)dx
o Variance of a Random Variable
Var(X) = E[X − E(X)]2 Var(X) = E[X − E(X)]2
↪→ where the second moment is obtained as follows:
E(X 2) =
n∑i=1
x2i · P[X = xi ] E(X 2) =
∫ ∞−∞
x2 · f (x)dx
o In addition, the following always holds:
Eg(X) =
n∑i=1
g(xi) · P[X = xi ] Eg(X) =
∫ ∞−∞
g(x) · f (x)dx
2 / 13STAT 235 | Lecture 4
N
Random Variable Characteristics
Random Variable Characteristicso Expected Value of a Random Variable
E(X) =
n∑i=1
xi · P[X = xi ] E(X) =
∫ ∞−∞
x · f (x)dx
o Variance of a Random Variable
Var(X) = E[X − E(X)]2 Var(X) = E[X − E(X)]2
↪→ where the second moment is obtained as follows:
E(X 2) =
n∑i=1
x2i · P[X = xi ] E(X 2) =
∫ ∞−∞
x2 · f (x)dx
o In addition, the following always holds:
Eg(X) =
n∑i=1
g(xi) · P[X = xi ] Eg(X) =
∫ ∞−∞
g(x) · f (x)dx
2 / 13STAT 235 | Lecture 4
N
Random Variable Characteristics
Random Variable Characteristicso Expected Value of a Random Variable
E(X) =
n∑i=1
xi · P[X = xi ] E(X) =
∫ ∞−∞
x · f (x)dx
o Variance of a Random Variable
Var(X) = E[X − E(X)]2 Var(X) = E[X − E(X)]2
↪→ where the second moment is obtained as follows:
E(X 2) =
n∑i=1
x2i · P[X = xi ] E(X 2) =
∫ ∞−∞
x2 · f (x)dx
o In addition, the following always holds:
Eg(X) =
n∑i=1
g(xi) · P[X = xi ] Eg(X) =
∫ ∞−∞
g(x) · f (x)dx
2 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o Jakob Bernoullio 1655 – 1705o one of the many prominent
mathematicians in theBernoulli family in Basel,Switzerland;
o he also discovered a very important constant...
limn→∞
(1+ 1
n
)n= e
3 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o Jakob Bernoullio 1655 – 1705o one of the many prominent
mathematicians in theBernoulli family in Basel,Switzerland;
o he also discovered a very important constant...
limn→∞
(1+ 1
n
)n= e
3 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o Jakob Bernoullio 1655 – 1705o one of the many prominent
mathematicians in theBernoulli family in Basel,Switzerland;
o he also discovered a very important constant...
limn→∞
(1+ 1
n
)n
= e
3 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o Jakob Bernoullio 1655 – 1705o one of the many prominent
mathematicians in theBernoulli family in Basel,Switzerland;
o he also discovered a very important constant...
limn→∞
(1+ 1
n
)n= e
3 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o the simplest construction of a random mechanism...o ...two possible outcomes only;
o typically, the outcomes are labeled with 1 (success) and 0 (fail);o however, other notation is easily applicable as well;
o the main characteristics (mean & variance):
EX = p VarX = p · (1− p)
↪→ for p ∈ [0, 1] being a probability of success;o What is the probability mass function and cumulative probability function?
4 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o the simplest construction of a random mechanism...o ...two possible outcomes only;
o typically, the outcomes are labeled with 1 (success) and 0 (fail);o however, other notation is easily applicable as well;
o the main characteristics (mean & variance):
EX = p VarX = p · (1− p)
↪→ for p ∈ [0, 1] being a probability of success;o What is the probability mass function and cumulative probability function?
4 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o the simplest construction of a random mechanism...o ...two possible outcomes only;
o typically, the outcomes are labeled with 1 (success) and 0 (fail);o however, other notation is easily applicable as well;
o the main characteristics (mean & variance):
EX = p VarX = p · (1− p)
↪→ for p ∈ [0, 1] being a probability of success;
o What is the probability mass function and cumulative probability function?
4 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Bernoulli Random Variables
o the simplest construction of a random mechanism...o ...two possible outcomes only;
o typically, the outcomes are labeled with 1 (success) and 0 (fail);o however, other notation is easily applicable as well;
o the main characteristics (mean & variance):
EX = p VarX = p · (1− p)
↪→ for p ∈ [0, 1] being a probability of success;o What is the probability mass function and cumulative probability function?
4 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?
=⇒ Binomial Distribution...=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...
=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
From Bernoulli to Others ...o it is the simplest but also the most important discrete distribution;↪→ many others are directly derived from it...
o Where to go next?=⇒ Binomial Distribution...
=⇒ Poisson Distribution...
=⇒ Geometric Distribution...=⇒ Negative Binomial Distribution...
o Discrete distributions are used to model random behavior of categoriesbut counts as well.
o Using counts, one has to possibly allow even for infinite number ofpossibilities...
5 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution
o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;
o independent repetitions of the Bernoulli trials: X ∼ B(n, p);
o What is the sample space? What is the mass probability function?What is the cumulative probability function?
o the main characteristics (mean & variance):
EX = np VarX = np(1− p)
↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;
6 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution
o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;o independent repetitions of the Bernoulli trials: X ∼ B(n, p);
o What is the sample space? What is the mass probability function?What is the cumulative probability function?
o the main characteristics (mean & variance):
EX = np VarX = np(1− p)
↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;
6 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution
o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;o independent repetitions of the Bernoulli trials: X ∼ B(n, p);
o What is the sample space? What is the mass probability function?What is the cumulative probability function?
o the main characteristics (mean & variance):
EX = np VarX = np(1− p)
↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;
6 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution
o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;o independent repetitions of the Bernoulli trials: X ∼ B(n, p);
o What is the sample space? What is the mass probability function?What is the cumulative probability function?
o the main characteristics (mean & variance):
EX = np VarX = np(1− p)
↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;
6 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution
o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;o independent repetitions of the Bernoulli trials: X ∼ B(n, p);
o What is the sample space? What is the mass probability function?What is the cumulative probability function?
o the main characteristics (mean & variance):
EX = np VarX = np(1− p)
↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;
6 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Binomial Distribution (PMS)
7 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;
o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distributiono it is defined as a number of total trials before the first success occurs;o given by one parameter only: p ∈ [0, 1] - the probability of success;o the random variable X ∼ G(p) can take any integer value (1, 2, 3, . . . );
o the main characteristics (mean & variance):
EX =1p VarX =
1− pp2
↪→ for p ∈ [0, 1] being a probability of the success;
o straightforward generalization into a random mechanism with moresuccesses... Negative Binomial Distribution
o ... the number of total trials before r ∈ N successes is achieved;o ... the main characteristics (mean & variance):
EX =rp VarX =
r · (1− p)p2
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes over the sequence of all trials;
8 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Geometric Distribution
9 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Negative Binomial Distribution
10 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
o random distribution used to describe random behavior of counts...(the number of 911 calls a day, the number of passengers in a train, etc.)
o discrete random distribution defined by one parameter only...X ∼ P(λ), for λ > 0
o ... the main characteristics (mean & variance):
EX = λ > 0 VarX = λ
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes in total;
o What is the mass distribution function? What is the cumulativedistribution function?
11 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
o random distribution used to describe random behavior of counts...(the number of 911 calls a day, the number of passengers in a train, etc.)
o discrete random distribution defined by one parameter only...X ∼ P(λ), for λ > 0
o ... the main characteristics (mean & variance):
EX = λ > 0 VarX = λ
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes in total;
o What is the mass distribution function? What is the cumulativedistribution function?
11 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
o random distribution used to describe random behavior of counts...(the number of 911 calls a day, the number of passengers in a train, etc.)
o discrete random distribution defined by one parameter only...X ∼ P(λ), for λ > 0
o ... the main characteristics (mean & variance):
EX = λ > 0 VarX = λ
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes in total;
o What is the mass distribution function? What is the cumulativedistribution function?
11 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
o random distribution used to describe random behavior of counts...(the number of 911 calls a day, the number of passengers in a train, etc.)
o discrete random distribution defined by one parameter only...X ∼ P(λ), for λ > 0
o ... the main characteristics (mean & variance):
EX = λ > 0 VarX = λ
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes in total;
o What is the mass distribution function? What is the cumulativedistribution function?
11 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
o random distribution used to describe random behavior of counts...(the number of 911 calls a day, the number of passengers in a train, etc.)
o discrete random distribution defined by one parameter only...X ∼ P(λ), for λ > 0
o ... the main characteristics (mean & variance):
EX = λ > 0 VarX = λ
↪→ for p ∈ [0, 1] being a probability of success and r ∈ N being thenumber of all successes in total;
o What is the mass distribution function? What is the cumulativedistribution function?
11 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
Poisson Distribution
12 / 13STAT 235 | Lecture 4
N
Discrete Random Variables
To be continued...
o Continuous random variables...o Continuous random variable characteristics;o Some most common continuous distributions;o Practical examples, simple transformations...o Some useful computational approaches...
13 / 13STAT 235 | Lecture 4
N
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