MATH 256 Probability and Random Processes

Yrd. Doç. Dr. Didem Kivanc Turelididemk@ieee.org

didem.kivanc@okan.edu.tr

14/10/2011 Lecture 3

OKAN UNIVERSITYFACULTY OF ENGINEERING AND ARCHITECTURE

04 Random Variables

Fall 2011

Probability Mass Function

• Is defined for a discrete variable X.

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p a P X a

0 1,2,...( ) 0

ip x ip a

p x x

for for all other values of

1

1ii

p x

• Suppose that

• Then since x must be one of the values xi,

Example of probability mass function

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0 0 1 4

1 1 1 2

2 2 1 4

p P X

p P X

p P X

Expectation of a random variable• If X is a discrete random variable having a probability mass

function p(x) then the expectation or the expected value of X denoted by E[X] is defined by

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: ( ) 0

( )x p x

E X xp x

• In other words, • Take every possible value for X• Multiply it by the probability of getting that value• Add the result.

Examples of expectation• For example, suppose you have a fair coin. You flip the coin,

and define a random variable X such that – If the coin lands heads, X = 1

– If the coin lands tails, X = 2

• Then the probability mass function of X is given by

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11 2

2p p 1 2 if 1 or 2,

0 otherwise.x x

p x Or we can write

1 11 2 1.5

2 2E X

Expectation of a function of a random variable

• To find E[g(x)], that is, the expectation of g(X)• Two step process: – find the pmf of g(x)– find E[g(x)]

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Let X denote a random variable that takes on any of the values –1, 0, and 1 with respective probabilities

{ 1} 0.2 { 0} 0.5 { 1} 0.3P X P X P X Compute 2E X

SolutionLet Y = X 2.

{ 1} { 1} { 1} 0.5{ 0} { 0} 0.5

P Y P X P XP Y P X

2 1(0.5) 0(0.5) 0.5E X E Y

Then the probability mass function of Y is given by

0.5 if 0 or 10 otherwise.

y yp y

Proposition 4.1

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If X is a discrete random variable that takes on one of the values xi, i ≥ 1 with respective probabilities p(xi), then any real valued function g.

i ii

E g X g x p x Check if this holds for the previous example:

22 2 21 0.2 0 0.5 1 0.3

1 0.2 0.3 0 0.50.5

E X

Proof of Proposition 4.1

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:

:

:

i j

i j

i j

i i i ii j i g x y

j ij i g x y

j ij i g x y

j jj

g x p x g x p x

y p x

y p x

y P g X y

E g X

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Variance

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Variance

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Some more comments about variance

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Bernoulli Random Variables

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Binomial Random Variables

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Poisson Random Variable

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Some examples of random variables that generally obey the Poisson probabilitylaw [that is, they obey Equation (7.1)] are as follows:1. The number of misprints on a page (or a group of pages) of a book2. The number of people in a community who survive to age 1003. The number of wrong telephone numbers that are dialed in a day4. The number of packages of dog biscuits sold in a particular store each day5. The number of customers entering a post office on a given day6. The number of vacancies occurring during a year in the federal judicial system7. The number of α-particles discharged in a fixed period of time from some radioactivematerial

Expectation of the sum of r.v.s• Start with this:

• It’s really quite easy to show that:

• And from this we show that

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