6More distributions
IA Probability
6.3 Beta distribution*
Suppose
X
1
, ··· , X
n
are iid
U
[0
,
1]. Let
Y
1
≤ Y
2
≤ ··· ≤ Y
n
be the order
statistics. Then the pdf of Y
i
is
f(y) =
n!
(i − 1)!(n − i)!
y
i−1
(1 − y)
n−i
.
Note that the leading term is the multinomial coefficient
n
i−1,1,n−1
. The formula
is obtained using the same technique for finding the pdf of order statistics.
This is the beta distribution: Y
i
∼ β(i, n −i + 1). In general
Definition (Beta distribution). The beta distribution β(a, b) has pdf
f(x; a, b) =
Γ(a + b)
Γ(a)Γ(b)
x
a−1
(1 − x)
b−1
for 0 ≤ x ≤ 1.
This has mean a/(a + b).
Its moment generating function is
m(θ) = 1 +
∞
X
k=1
k−1
Y
r=0
a + r
a + b + r
!
θ
k
k!
,
which is horrendous!