Understanding Random Variables: Concept, Distribution Functions, and Mean Values, Assignments of Statistics

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RANDOM VARIABLES

OUTLINE

Concept of A Random Variable

Distribution Functions

Density Functions

Mean Values and Moments

Reading:

G. R. Cooper & C. D. McGillem 2.1 - 2.

EE/STAT 322, #

CONCEPT OF A RANDOM VARIABLE

space, or, a function mapping sample space into the real line. Mathematically: An assignment of a real number to each point in sample

Random variable X

Sample Space

Ω

Real number line

x

Idea

Randomness is in the sample space and probability assignment.

A

of the experiment yields a specific sample pointrandom variable just assigns a number to each sample point. A performance

ω

, which produces a sample

value, say

x = X ( ω )

, of the random variable.

EE/STAT 322, #

CONCEPT OF A RANDOM VARIABLE (CONT.)

Function:

x ( t ) , where the whole set of

t

is called the

domain

of function

x ( t ) , and the set of

x

is called the

range

of the function.

For example,

x ( t ) =

t 2

is a function, where

t

, and

x

Definition of a random variable (RV): an RV

x

is a process of assigning

a number

x ( ξ )

to every outcome

ξ .

• { X ≤ x }

represents a subset of

S

consisting of all outcomes

ξ

such

that

x ( ξ ) ≤ x.

Thus

X

x }

is not a set of numbers but a set of

experimental outcomes.

EE/STAT 322, #

CONCEPT OF A RANDOM VARIABLE (CONT.)

Example:

Define an RV

X

f i ) = 10

i

for the die experiment, where

i

is the number

of the die.

Thus,

X

f 1 ) = 10

, X

f 2 ) = 20

,... , X

f 6 ) = 60

The set

X

consists of elements

f 1 , f

2 , f

3

only.

The set

X

consists of

f 2 , · · ·

, f

6 .

Alternatively, if we defined

X

f 1 ) =

X

f 3 ) =

X

f 5 ) = 0

, and

X

f 2 ) =

X

f 4 ) =

X

f 6 ) = 2

Then set

{ X ≤ 1 } = { f 1

, f

3 , f

5 } ,

and set

{ X ≥ 2 } = { f 2

, f

4 , f

6 } .

EE/STAT 322, #

DISTRIBUTION FUNCTIONS

Definition:

the probability distribution function of the RV

X

is

F

X

(^) ( x ) =

P

X

x } ,

where

x

Example:

In the fair die experiment, define an RV

X

such that

x ( f i ) =

i ,

the distribution function of

X

is then a staircase function

P

X

x ) =

b x c / 6 0 ≤

x <

x <

x

For

example,

F

P

f 1 , f

2 , f

3 , f

and

F

P

f 1 , (^) · · ·

, f

6 ) = 1

Notation

: upper case letters for RVs and lower ones for their values.

EE/STAT 322, #

7

PROPERTIES OF A DISTRIBUTION FUNCTION

First, we define

F

x

)

and

F

x −

)

as

F

x

) = lim

ε →

0 F

(^) ( x

ε )

and

F

x −

) = lim

ε →

0 F

(^) ( x

ε ) .

0 ≤ F x ( x ) ≤ 1 ,

< x <

F

and

F

Because

F

P

X

P

S

F

P

3. If

x 1

< x

2 , then

F

x 1 )

≤

F

x 2 ) .

Because

P

X ≤ x 2 } = P

X ≤ x 1 } + P

x 1

< X

x 2 } .

P

x 1

< X

x 2 ) =

F

x 2 )

−

F

x 1 ) .

Because

P

x 1

< X

x 2 ) =

P

X ≤ x 2 ) − F

X ≤ x 1 ).

EE/STAT 322, #

DENSITY FUNCTION

density function (pdf), as shown by Definition: the derivative of the distribution function is called the probability

f (^) ( x ) =

d F

x )

d x

lim

∆

x →

0

F

(^) ( x

x )

−

F

x )

x

f The integral of pdf relates to the distribution function. x ( x ) dx

P

x < X

x

dx

f x ( x ) ≥ 0 ,

< x <

∞ −∞

f x ( x ) dx

F

P

S

F

x ( x ) =

x −∞

f x ( u ) du

x 2

x 1 f x ( u )

du

F

x 2 )

−

F

x 1 ) =

P

x 1

< X

x 2 ) .

EE/STAT 322, #

PROBABILITY MASS FUNCTIONS

Abbreviation: PMF; Suitable for description of discrete RVs

For continuous RVs, will use Probability Density Function (pdf).

Definition and Notation

p X

(^) ( x ) =

P

X

x }

Random variable X )

Sample Space

Ω

Real number line

x

0

1

Probability Law

EE/STAT 322, #

DENSITY FUNCTIONS

Example:

Let

X

be the number of heads obtained from two independent

p tosses of a fair coin X

(^) ( x ) =

if

x

or

x

if

x

otherwise

x

p

X

(x)

EE/STAT 322, #

REMARKS ABOUT PMF

• PMF PROPERTIES

p X

(^) ( x )

≤

for all

x

x

p X

(^) ( x ) = 1

say that a random variable is a Random variables are often referred to according to their PMF’s. E.g., we

Bernoulli RV

if its PMF is

p X

(^) ( x ) =

p,

if

x

p,

if

x

EE/STAT 322, #

Example:

Y

X

2 , and

X

has a density

f X

(^) ( x )

. Determine the distribution

function and PDF of

Y

If Solution:

y

, then

x 2

≤

y

corresponds to

− √ y ≤ x ≤ √ y

, and

F

Y

( y ) =

P

√ y ≤ x ≤ √ y } = F X

√ y ) − F X

y ) ,

If

y <

0 , x 2 ≤ y

has no solution for a real RV

x , and

F

Y

( y ) =

P

Differentiating

F

Y

( y ) , we get

f Y

( y ) =

1

2 √

y (^) ( f X

(^) ( √

y ) +

f X

(^) ( −

√

y ))

y

≥

y <

EE/STAT 322, #

MEAN AND MOMENTS

Mean of RV

X

E

X

−∞∞

xf

x ) dx

Mean of function

g ( X

1. Method I:

E

[

g ( x )] =

∞ −∞

g ( x ) f (^) ( g ( x ))

dg

x ) ;

2. Method II:

E

[

g ( X

)] =

∞ −∞

g ( x ) f X

x ) dx

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 )

EE/STAT 322, #

MEAN AND MOMENTS (CONT.)

Linearity of the expectation

E [ g 1 ( X

g 2 ( X

)] =

E

[

g 1 ( X

)] +

E [ g 2 ( X

)]

• E [ X 1 + X 2 +

X

m

] =

E

[

X

1 ] +

E

[

X

2 ] +

• E [ X m ] ,

• E [ g 1 ( X

g 2 ( X

g m

( X

)] =

E [ g 1 ( X

)]+

E [ g 2 ( X

)]+

• E [ g m ( X

)]

But generally

E [ X 1 X 2 ] 6

= E [ X 1 ]

E [ X 2 ].

EE/STAT 322, #

19

MEAN AND MOMENTS (CONT.)

Variance var

X

σ x 2

=

E

[(

X

E

[

X

])

2 ] =

E

[

X 2 − 2 X ¯

X

X

2 ]

E

[

X 2 ] − 2 E [

X

] ¯

X

X 2 = E [ X 2 ] −

X

2

• ¯

X

2

E

[

X

2 ] −

X 2 = E [ X 2 ] − ( E [ X

])

2

Moments:

E

[

X

n ] =

∑ x x n p X

x )

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

EE/STAT 322, #