Lecture 16
16.1
Fisher and Student distributions.
Consider X1 , . . . , Xk and Y1 , . . . , Ym all independent standard normal r.v.
Definition: Distribution of the random variable
Z=
X12 + . . . + Xk2
Y12 + . . . + Ym2
is called Fisher distribution with degree of freedom k and m, and it is denoted as
Fk,m .
Let us compute the p.d.f. of Z. By definition, the random variables
X = X12 + . . . + Xk2 ∼ χ2k and Y = Y12 + . . . + Ym2 ∼ χ2m
have χ2 distribution with k and m degrees of freedom
Recall that χ2k
correspondingly.
distribution is the same as gamma distribution Γ k2 , 12 which means that we know
the p.d.f. of X and Y :
k
m
(1)2 k
(1) 2 m
1
1
X has p.d.f. f (x) = 2 k x 2 −1 e− 2 x and Y has p.d.f. g(y) = 2 m y 2 −1 e− 2 y ,
Γ( 2 )
Γ( 2 )
for x ≥ 0 and y ≥ 0. To find the p.d.f of the ratio X
, let us first recall how to write
Y
its cumulative distribution function. Since X and Y are always positive, their ratio
is also positive and, therefore, for t ≥ 0 we can write:
X
(X ≤ tY ) = {I(X ≤ tY )}
≤t =
Y
Z ∞Z ∞
=
I(x ≤ ty)f (x)g(y)dxdy
0
0
Z ∞ Z ty
=
f (x)g(y)dx dy
0
0
60
61
LECTURE 16.
Y
X=tY
Y0
X<=tY0
X
Figure 16.1: Cumulative Distribution Function.
where f (x)g(y) is the joint density of X, Y. Since we integrate over the set {x ≤ ty}
the limits of integration for x vary from 0 to ty (see also figure 16.1).
Since p.d.f. is the derivative of c.d.f., the p.d.f. of the ratio X/Y can be computed
as follows:
Z Z
Z ∞
d X
d ∞ ty
≤t =
f (x)g(y)dxdy =
f (ty)g(y)ydy
dt
Y
dt 0
0
0
m
Z ∞ 1 k
( 12 ) 2 m −1 − 1 y
(2)2
k
1
−1
−
ty
(ty) 2 e 2
y 2 e 2 ydy
=
Γ( m2 )
Γ( k2 )
0
k+m
Z ∞
( 21 ) 2
k
−1
( k+m
)−1 − 12 (t+1)y
2
2
=
y
e
dy
t
Γ( k2 )Γ( m2 )
|0
{z
}
The function in the underbraced integral almost looks like a p.d.f. of gamma distribution Γ(α, β) with parameters α = (k + m)/2 and β = 1/2, only the constant in
front is missing. If we miltiply and divide by this constant, we will get that,
k+m
k+m
Z ∞ 1
)
( 21 ) 2
Γ( k+m
( 2 (t + 1)) 2 ( k+m )−1 − 1 (t+1)y
k
d X
−1
2
2
t
y 2
≤t =
e 2
dy
k+m
dt
Y
)
Γ( k2 )Γ( m2 )
Γ( k+m
( 21 (t + 1)) 2 0
2
=
Γ( k+m
) k −1
k+m
2
t 2 (1 + t) 2 ,
k
m
Γ( 2 )Γ( 2 )
since we integrate a p.d.f. and it integrates to 1.
To summarize, we proved that the p.d.f. of Fisher distribution with k and m
degrees of freedom is given by
Γ( k+m
) k
k+m
fk,m (t) = k 2 m t 2 −1 (1 + t)− 2 .
Γ( 2 )Γ( 2 )
62
LECTURE 16.
Next we consider the following
Definition. The distribution of the random variable
Z=q
X1
1
(Y12
m
+ · · · + Ym2 )
is called the Student distribution or t-distribution with m degrees of freedom and it
is denoted as tm .
Let us compute the p.d.f. of Z. First, we can write,
(−t ≤ Z ≤ t) =
(Z 2 ≤ t2 ) =
t2 X12
≤
.
Y12 + · · · + Ym2
m
If fZ (x) denotes the p.d.f. of Z then the left hand side can be written as
Z t
fZ (x)dx.
(−t ≤ Z ≤ t) =
−t
X2
1
On the other hand, by definition, Y 2 +...+Y
2 has Fisher distribution F1,m with 1 and
m
1
m degrees of freedom and, therefore, the right hand side can be written as
Z
t2
m
f1,m (x)dx.
0
We get that,
Z
t
fZ (x)dx =
−t
Z
t2
m
f1,m (x)dx.
0
Taking derivative of both side with respect to t gives
t2 2t
fZ (t) + fZ (−t) = f1,m ( ) .
m m
But fZ (t) = fZ (−t) since the distribution of Z is obviously symmetric, because the
numerator X has symmetric distribution N (0, 1). This, finally, proves that
fZ (t) =
Γ( m+1
) t2 −1/2 t2 − m+1
) 1
t
t2
t Γ( m+1
t2 − m+1
2
2
2
√
=
f1,m ( ) =
1+
) 2 .
(1+
m
m
m Γ( 21 )Γ( m2 ) m
m
m
Γ( 12 )Γ( m2 ) m