chapter 3: expectations

7/29/2019 Chapter 3: Expectations

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Chapter 3: Expectations

3.1 Introduction

Traditionally -> get involved in betting and

other games of chance with monetary reward

Thus, mathematical expectation is AVERAGE

AMOUNT you can get when participating in suchgames

Example: Roll a die and receive the money

according to the number that shows up 1 => $1; 2 => $2; 6 => $6

What is the average amount one can get if this game is

repeated in a long run?

1 NAA, Universiti Pendidikan Sultan Idris, Sem 1, 2011/12


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Example: Roll a die and receive the money

according to the number(continued) Suppose the die is fair, then the law of large number

suggests that the six sides will turn up about equally

often.

Thus, mathematical expectation is just:

If you play this game long enough, you expect to get

$3.50. Why this information is important in this game of

chance?


50.3$6

16$

6

15$

6

14$

6

13$

6

12$

6

11$


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3.2 Expected values

Definition: Let X => a discrete r.v. defined on ,

and its possible values are defined x1, x2, with

the corresponding probabilities f(x1), f(x2),

Thus, the expected value for X is:

Other notations used:

X EX


Probability of

each xi

Value of each

xi


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Suppose that X is a discrete r.v. and we have a

function, Y = g(X)

Because Y is a function of X => Y is also a r.v.

Whenever we have a r.v., we can find its expectation

The formula is:


Similar concept to what weve discussed

Value of function g(xi)

Probability

for each xi


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Example: An experiment consists of tossing 3

coins. A random variable X represents the

number of heads. Find what is the expected

value of X.



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Example: Suppose a random variable X can take

values {1, 2, 3} and its corresponding

probabilities are given as 0.1, 0.5 and 0.4. If a

function Y is defined as Y = X2 + 2X, find the

mean value of Y. Solution:


Note that probabilities are already given, so

we are left to find the values of the function


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If X and Y are discrete r.v., then:



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Example: Suppose X denotes the number of

heads appearing in 3 coin tosses and E(X) is

given as 3/2. Suppose a function Y is defined as

Y = 2X + 3. Find the expected value of Y.

Solution:



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Example: Suppose X and Y are two discrete

random variables and its joint p.f. is illustrated

in the following table. Find the expected value

of X and Y; that is E(X + Y).


1 2 3

1 1/12 1/6 1/122 1/6 0 1/6

3 0 1/3 0

Y

X


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Definition: If X is a continuous r.v. with p.d.f.

f(x), the mean of X or the expected value of X:

If the interest is on a function Y = g(X), then the

mean value or the expected value of Y is:



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Example: Find the mean value of X when X has

the following p.d.f. f(x) = 6x (1-x) for 0


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Example: Suppose X is uniform on interval (-1,

1) and define Y = X2. Find the mean value of Y.

Solution:

If X is uniform, then p.d.f. of X is just 1/2. How do you get

this result? Next, find E(Y)



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3.3 Functions of random variables

Suppose we have a random vector (X, Y) and its

p.d.f. f(x, y). Then the expected value of a

function g(X, Y) is just:

If X and Y have expected values, then forconstants a, b and c:



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Example: Suppose the joint p.d.f. for random

vector (X, Y) is f(x,y) = 24xy for x>0, y>0, x+y


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Example: Suppose we have the following

random variables, X1, X2,, Xn and the expectedvalues for each Xi is . Suppose function Y is

defined as the sum of X1, X2,Xn. Find the

expected value of Y? Solution:



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Two random variables can be independent of

each other.

Suppose X and Y are independent, then

Example: If E(X) is given as 9 and E(Y) is 3/2 and

events X and Y are independent, find the mean

value of E(XY).



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3.4 Moments

Concept of E(X) => describes the middle or

center of the distribution

Concept can be extended to the followings:

Expected value of integer powers of X

Powers of deviations about a particular value


Moments'

k

kXE

kth moment

k

k

XE

kth central

moment


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What is the first moment of X?

What is the second moment of X?

What is the first central moment of X?

What is the second central moment of X?



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3.4 Variance

Definition: Variance is the second central

moment:

If X is a discrete r.v., then var X can be obtained

by implementing the expectation concept

Recall:


E[ (X - )2 ]


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If X is continuous r.v. then:

It is always easy, however to calculate varianceX using this formula:


22

22

222

2

2

XE

XEXE

XXEXE

How to get this???


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Prove that for any constant a,

Solution:


22 aXEaXEXX

22XX

aXEaXE

222 2XXXX

aaXXEaXE

222 2XXXX

aXEaXEaXE


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Example: Suppose that the p.f. is given as f(x) =

1/6 for x = 1, 2,6 and the mean of X is = 7/2.Find the standard deviation of r.v. X

Solution:



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Example: Suppose that X is uniform on the

interval (0, b). What is the p.d.f. of X? Find thevariance of X.

Solution:



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3.5 Chebyshev inequality

For any constant c > 0, and if X has mean and

standard deviation , then

The equivalence form is



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Interested in: finding the probability that X is

greater than kstandard deviations from themean:

What is the probability X is greater than 2 standard

deviations from the mean?

What is the probability X is greater than 3 standard

deviations from the mean?


22

2

kkXP


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Example: Consider the discrete r.v. X defined by

P(X=0) = 3/4; P(X=a) = P(X=-a) = 1/8. Find theexpectation of X and the variance of X. Next,

find what is the probability X is greater than 2

standard deviations from the mean. Solution:



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3.6 Covariance

The concept of expectation can be extended to

find the covariance.

Recall that variance X is

Now, covariance is a term used to describe the

association of 2 variables, say X and Y.


Positive: X is large when Y is large

OR X is small when Y is small

Negative: X is large when Y is

small OR vice versa


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Some other useful results:

cov (X, X) is just variance of X => how to prove this?

Given some constants a and b, then:

If X and Y are independent, then



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3.7 Correlation

Covariance suffers from being dependent on

the units of measurement

Example: Suppose X denotes the weight of a person

and Y denotes the height of a person, so the unitmeasurement in covariance involves say kg and cm.

Eliminate the problem => correlation coefficient

Also measures the association between 2 variables But unit-less


Can you see why

correlation is unit-less?


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3.7 Correlation

Correlation is 0 when covariance is 0. Can easily

see this from the formula.

Value of ranges from -1 to 1.

Highly correlated when the values are close to

either -1 or 1

Positive correlation and negative correlation

Follows the same definition of +ve andve covariance


The variables are uncorrelated

X and Y independent

NOT TRUE


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Example: Suppose a random vector X and Y are

described by the joint p.f. given as follows. Findthe covariance of (X, Y). Next find the

correlation of (X, Y).


1 2 3 4

1 0 1/12 1/12 1/12

2 1 2 3 4

3 1/12 1/12 0 1/12

4 1/12 1/12 1/12 0

X

Y


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Example: Suppose X, Y and Z are i.i.d

(independent and identically distributed)random variables with mean 0 and st. deviation

1. Find var (2X + 3Y Z). Next, find cov (X 2Y,

3X + Y + 2Z).

32 NAA Universiti Pendidikan Sultan Idris Sem 1 2011/12

chapter 3: expectations

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