intro stats for engineering class stat 235 -...

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

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

E(X) =

n∑i=1

xi · P[X = xi ] E(X) =

∫ ∞−∞

x · f (x)dx

E(X 2) =

n∑i=1

x2i · P[X = xi ] E(X 2) =

∫ ∞−∞

x2 · f (x)dx

Eg(X) =

n∑i=1

g(xi) · P[X = xi ] Eg(X) =

∫ ∞−∞

g(x) · f (x)dx

E(X) =

n∑i=1

xi · P[X = xi ] E(X) =

∫ ∞−∞

x · f (x)dx

E(X 2) =

n∑i=1

x2i · P[X = xi ] E(X 2) =

∫ ∞−∞

x2 · f (x)dx

Eg(X) =

n∑i=1

g(xi) · P[X = xi ] Eg(X) =

∫ ∞−∞

g(x) · f (x)dx

E(X) =

n∑i=1

xi · P[X = xi ] E(X) =

∫ ∞−∞

x · f (x)dx

E(X 2) =

n∑i=1

x2i · P[X = xi ] E(X 2) =

∫ ∞−∞

x2 · f (x)dx

Eg(X) =

n∑i=1

g(xi) · P[X = xi ] Eg(X) =

∫ ∞−∞

g(x) · f (x)dx

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→∞

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?

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?

EX = p VarX = p · (1− p)

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...

o Where to go next?

=⇒ Binomial Distribution...=⇒ Poisson Distribution...

=⇒ Geometric Distribution...

=⇒ Negative Binomial Distribution...

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?

EX = np VarX = np(1− p)

↪→ for p ∈ [0, 1] being a probability of success and n ∈ N;

o discrete distribution given by two parameters: p ∈ [0, 1] and n ∈ N;o independent repetitions of the Bernoulli trials: X ∼ B(n, p);

Binomial Distribution (PMS)

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, . . . );

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;

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, . . . );

EX =1p VarX =

1− pp2

EX =rp VarX =

r · (1− p)p2

EX =1p VarX =

1− pp2

EX =rp VarX =

r · (1− p)p2

EX =1p VarX =

1− pp2

EX =rp VarX =

r · (1− p)p2

EX =1p VarX =

1− pp2

EX =rp VarX =

r · (1− p)p2

EX =1p VarX =

1− pp2

EX =rp VarX =

r · (1− p)p2

Geometric Distribution

Negative Binomial Distribution

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?

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...

intro stats for engineering class stat 235 -...

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