Functions of Random Variables: Expectation, Mean, and Variance, Study notes of Statistics

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EXPECTATION, MEAN, VARIANCE

OUTLINE

Review of Concepts of Random Variables and PMF

Functions of Random Variables

Expectation, Mean, Variance

Reading

: Bertsekas & Tsitsiklis, 2.3, 2.

EE/STAT 322, #

FUNCTIONS OF RANDOM VARIABLES

Suppose we have a RV

X

defined on an experiment, i.e.

a function

X

ω

) .

Y Furthermore, suppose that we have a function of the random variable

g ( X

, such as

X 2 , | X |

cos

( X ) , e 3 X

.^

byThen, these are also random variables defined on the original experiment

Y

ω

) =

g ( X

ω

))

, i.e. a function of a function is just another function.

Since

Y

is just a random variable, it must have a PMF,

p Y

( y ) , which we

can find from the original probability model.

There is a more direct and usually simpler way: find PMF for

Y = g ( X )

from the PMF of

X

EXAMPLE

X

= maximum of two rolls of a four-sided die

• Y = g ( X )

as

g ( x ) =

{ 1 , x ≥ 3

0 ,

otherwise

Find the PMF of

p Y Y

( y ) =

x : g ( x )=

y p X

(^) ( x )

p Y

(1) =

x : x ≥

3 p X

(^) ( x ) =

p Y

(0) = 1

p Y

(1) =

EXPECTATION

Expectation

The

expected value

expectation

or

mean

of a random variable

X

with PMF

p X

(^) , is defined as

E

[

X

] =

x

xp

X

(^) ( x )

c = mean = E[X] center of gravity

x

EXPECTED VALUE RULE FOR FUNCTIONS OF

RANDOM VARIABLES

Let

Y = g ( X )

be a function of the RV

X

E

[

Y

] =

E

[

g ( X

)] =

∑ x g ( x ) p X

x )

E

[

g ( X

)] =

E

[

Y

] =

y

yp

Y

( y ) =

y

y

{ x | g ( x )=

y } p X

(^) ( x )

y

{ x

| g ( x )=

y }

yp

X

(^) ( x ) =

y

{ x | g ( x )=

y } g ( x ) p X

x ) =

∑ x g ( x ) p X

x )

EXPECTED VALUE RULE (CONT’D)

Linearity of the expectation

E [ g 1 ( X

g 2 ( X

)] =

E

[

g 1 ( X

)] +

E [ g 2 ( X

)]

Variance: var

X

E

[(

X

E

[

X

])

2 ] =

∑ x ( x − E [

X

])

2 p X

(^) ( x )

Moments:

E

[

X

n ] =

∑ x x n p X

x )

Variance in terms of moments: var

X

E [ X 2 ] − ( E [ X

])

2

Mean and variance of a linear function of a RV

Let

Y

aX

b

E

[

Y

] =

aE

[

X

] +

b,

var

Y

a

2 var

X