Chris Morgan, MATH G160csmorgan@purdue.edu

February 3, 2012Lecture 11

Chapter 5.3: Expectation (Mean) and Variance

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Expected Value

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Question: How do you determine the “value” of a game? Is it better to play Roulette than the Lottery? We are looking for ways of describing random variables.

Definition of an Expected Value

–The expected value of a random variable X with PMF is given by:

–The expected value is a weighted average of the possible values of X, weighted by the probabilities.

( ) * ( )E X x p x

Expected Value

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We may interchangeably use the terms mean, average, expectation, and expected value and the notations E(X) or μ

Note: The expected value of a random variable can be understood as the long-run-average value of the random variable in repeated independent trials. If you are playing a game, and X is what you win in the game, then E(X) would be your average win if you would play the game many many many many many many many many many times.

Example #1

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Recall from last lecture:

x P(x)0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

1 4 6 4 1( ) 0 1 2 3 4

16 16 16 16 16E X

Fundamental Expected-Value Formula

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Instead of E(X) we can also compute the expected value of a function of X.

– If X is a discrete random variable with PMF and is any real valued function of X, then:

[ ( )] ( )* ( )E g x g x p x

Example #2

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Recall from last lecture:

I can also compute:

x P(x)0 1/16 1 4/16 2 6/16 3 4/16 4 1/16

1 4 6 4 1( ) 0 1 2 3 4 2

16 16 16 16 16E X

2 2 2 2 2 21 4 6 4 1( ) 0 1 2 3 4 5

16 16 16 16 16E X

Example #3

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Or I can compute:

or even:

x P(x)0 1/16 1 4/16 2 6/16 3 4/16 4 1/16

1 4 6 4 1( 3) (0 3) (1 3) (2 3) (3 3) (4 3) 5

16 16 16 16 16E X

1 4 6 4 1(2 ) (2*0) (2*1) (2*2) (2*3) (2*4) 4

16 16 16 16 16E X

So then:

E(X+3) = E(X) + 3E(2X) = 2*E(X)

E(X2) ≠ E(X)2

Expectation in a Linear Operator

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Let X be a random variable and a, b be constants. Then:

Let X1,…,Xn be random variables. Then:

( ) ( )E aX b aE X b

1 1

( )n n

i ii i

E X E X

( )E b b (10) 10E

Variance

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The definition of variance of a random variable is a measure of the spread of its distribution. It is the expected squared deviation from the mean:

where μ = E[X]

If we know the pmf of X then we can calculate the variance as follows:

We can simplify the variance equation to this:

2( ) ( ) * ( )Var X k f k

2 2 2( ) [( ) ] ( ) [ ( )]Var X E X E X E X

2 2( ) [( ) ]x Var x E X

Variance

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- Var(X) is always non-negative (Var(X) >= 0)

- Sometimes, we’ll abbreviate: σ2

- Var(X) is a measure of the spread of the random variable. If Var(X)=0, then the spread is zero, i.e. all the probability is concentrated in one point (nothing is random anymore).

- The variance is not measured in the same units that the random variable is measure in. (This is a disadvantage!)

Example #4

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Recall from last lecture:

I can also compute:

Then:

1 4 6 4 1( ) 0 1 2 3 4 2

16 16 16 16 16E X

2 2 2 2 2 21 4 6 4 1( ) 0 1 2 3 4 5

16 16 16 16 16E X

2 2 2( ) ( ) [ ( )] 5 [2] 1Var X E X E X

Example #4

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Theorem: Variance is not a linear operator! Let X be a random variable and a, b, c be constants. Then:

2( ) ( )Var aX b a Var X

( ) 0Var c

Variance

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If X1, X2, …, Xn are independent, then:

1 1

( )n n

i ii i

Var X Var X

Standard Deviation

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Standard deviation of a random variable X is:

Note: unlike variance, standard deviation is measured in the same units

( ) ( )SD X Var X

Practice #5

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Let X be a discrete random variable with PMF:

E(X)=

Var(X)=

E(2X-3)=

Var(2x-3)=

X 0 1 2 3

p(x) 0.4 0.2 0.3 0.1

Practice #6

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Let X be a RV with mean μ=5 and variance σ²=9

Find E((X-1)2).

Find the standard deviation of X.

Practice #7

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For a game, you tell a friend that if a 6-sided die rolls a 2, you will pay her $2. If the die rolls a 3, she will pay you $3. Any other numbers (so 1, 4, 5, or 6) you pay her a quarter.

Let W be the random variable representing your friend’s winnings.

Practice #7

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What is the pmf of W?

What is the expected amount of money your friend will win?

Practice #7

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What is the standard deviation of your friend’s winnings?

Practice #7

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If you and your friend played this game 5 times, what would the overall expected value and standard deviation of your friend’s winnings be?

Practice #8

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If Var(Z) = 4, then find:

Var(5) =

Var(Z+1) =

Var(2Z) =

Var(aZ + b) =

Var(b-aZ) =

Practice #8

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Given that the Var(Y) = 9 and the E(Y) = 4, can we find E(Y2)?