3Linear models

IB Statistics

3.4 The F distribution

Definition (F distribution). Suppose U and V are independent with U ∼ χ

2

m

and

V ∼ χ

2

n

. The

X

=

U/m

V/n

is said to have an

F

-distribution on

m

and

n

degrees

of freedom. We write X ∼ F

m,n

Since

U

and

V

have mean

m

and

n

respectively,

U/m

and

V/n

are approxi-

mately 1. So F is often approximately 1.

It should be very clear from definition that

Proposition. If X ∼ F

m,n

, then 1/X ∼ F

n,m

.

We write

F

m,n

(

α

) be the upper 100

α

% point for the

F

m,n

-distribution so

that if X ∼ F

m,n

, then P(X > F

m,n

(α)) = α.

Suppose that we have the upper 5% point for all

F

n,m

. Using these in-

formation, it is easy to find the lower 5% point for

F

m,n

since we know that

P

(

F

m,n

<

1

/x

) =

P

(

F

n,m

> x

), which is where the above proposition comes

useful.

Note that it is immediate from definitions of

t

n

and

F

1,n

that if

Y ∼ t

n

, then

Y

2

∼ F

1,n

, i.e. it is a ratio of independent χ

2

1

and χ

2

n

variables.