Distributions Arising from the Normal

Chi-Square and F-Distributions

Instructor: Dr. Gaurav Bhatnagar

22001: PROBABILITY AND STATISTICS

Chi-square Distribution

A random variable is having Chi-square distribution if itcan be written as the sum of squares of mutual independentstandard normal random variables, i.e.,

All

The number of standard normal random variable is calledthe degree of freedom of the random variable.

Notation:

X

2 2 2 21 2 3 ... (1)nX Z Z Z Z

~ (0,1)iZ N

2~ nX

PDF of Chi-square Distribution

Given:

Let be the MGF of X, then

where

2 2 2 21 2 3 ... nX Z Z Z Z

2 22

1

( ) ( ) ( ) ( )i i

n n

X Z ZZ i

M t M t M t M t

( )XM t

2 2

2

21

1 2 /2 2

( ) ( )

1 11 2 ,22

i i

i

i

t z t zi iZ

t zi

M t E e e f z dz

e dz t t

22

1( ) ( ) 1 2 , (2)2i

nn

X ZM t M t t t

PDF of Chi-square Distribution

Recall the MGF of , which is given by

Comparing Eqns. (2) and (3)

~ ( , )Y

( ) (3)YM tt

1( ) ( ) iff and2 2

1~ ,2 2

X YnM t M t

nX

22( ) ( ) 1 2 (2)

i

nn

X ZM t M t t

PDF of Chi-square Distribution

Since

Then the pdf of X is given by

1~ ,2 2nX

12

212 2( ) , 0 (4)

( / 2)

nx xe

f x xn

A random variable has the Chi-square distribution with

n degree of freedom if its pdf is given by Eqn. (4).X

PDF of Chi-square Distribution

Chi-square Distribution

'

12 2

2

( ) ( )

1

2 ( / 2)

r rr

n xr

n

E X x f x dx

x e dxn

Moments:

Mean: r = 1

Variance:

' 2 2

2

1

2 ( / 2)

n x

r n x e dx nn

2' 2 2( ) ( ) ( 2) 2r E X E X n n n n

Chi-square Distribution

(1 2 )12 2

02

2

( ) ( ) ( )

1

2 ( / 2)

1(1 2 ) ,2

xt xtX

n t x

n

n

M t E e e f x dx

x e dxn

t t

Moments Generating Function:

Let and be two independent Chi-square random

variables with degree of freedom n and m, respectively.Then the sum is also a Chi-square random variable with

n+m degree of freedom.

Proof: The proof is straightforward using MGF.

1X

1 2 1 2

/2 /2 ( )/2

21 2

( ) ( ) ( )

(1 2 ) (1 2 ) (1 2 )~

X X X X

n m n m

n m

M t M t M t

t t tX X

Chi-square Distribution

2X

Probabilities with Chi-square distribution: Let be anyvalue between (0,1). Then there exist such that

2,nP X

Chi-square Distribution

2,n

2,n

Find 20.01,21 0.01P X

Chi-square Distribution20.01,21

F Distribution

A random variable is having F distribution if it can bewritten as the ratio of two mutual independent Chi-squarerandom variables, i.e.,

where and

The RV X is said to be of degree of freedom m and n.

Notation:

X

1

2

/ (6)/

Q mXQ n

21 ~ mQ

,~ m nX F

22 ~ nQ

PDF of F Distribution

21

2

2

( ) , 0 (7), 12 2

m

m

m n

mxnf x x

m n mB xn

A random variable has the F distribution with m and ndegree of freedom if its pdf is given by following equation.

X

F Distribution

Mean:

Variance:

Moment Generation Function: Does not exist (Check!!!)

'1 , 2

2n n

n

222

22

2

2

( ) ( )

( 2)( 2)( 4) 2

2 ( 2) , 4( 2) ( 4)

E X E X

n m nm n n n

n n m nm n n

Probabilities with F distribution: Let be any value between(0,1). Then there exist such that

, ,m nP X F

F Distribution

, ,m nF

, ,m nF

Important relation:

1, , , ,

2

2 2

1 , , 1 , ,

2

1 , ,

//

/ /1 11/ /

/ 1 1 (8)/

m n m n

m n m n

m n

Q mP X F P FQ n

Q n Q nP PQ m F Q m F

Q nPQ m F

F Distribution

Important relation (Contd…):

From Eqns. (8) and (9), we have

21 , ,

1

/1 (9)/ m n

Q nP FQ m

F Distribution

1 , ,, ,

1 (10)m nm n

FF

This relation says:

F Distribution

0.9,5,70.1,7,5

1 1 0.29673.37

FF