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.