Page 232 - Elements of Distribution Theory

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052184472Xc07 CUNY148/Severini May 24, 2005 3:59

218 Distribution Theory for Functions of Random Variables

When the distribution given by F is either absolutely continuous or discrete, it is possible
to derive the density function or frequency function, respectively, of the distribution.

Theorem 7.7. Let X 1 , X 2 ,..., X n denote independent, identically distributed, real-valued
random variables each with distribution function F and range X.
(i) If X 1 has a discrete distribution with frequency function p, then X (m) has a discrete
distribution with frequency function
m n+k−m
n m−k n − m + k j n−m+k− j
p (m) (t) = F(t 0 ) p(t) [1 − F(t)] ,
m − k j
k=1 j=k
for t ∈ X; here t 0 is the largest element of X less than t.
(ii) If X 1 has an absolutely continuous distribution with density function p, then X (m)
has an absolutely continuous distribution with density function
n − 1 m−1 n−m

p (m) (t) = n F(t) [1 − F(t)] p(t), −∞ < t < ∞.
m − 1
Proof. First consider the case in which X 1 has a discrete distribution. Let t denote a fixed
element of X. Each observation X 1 ,..., X n falls into one of three sets: (−∞, t 0 ], {t},or
[t 1 , ∞). Here t 0 denotes the largest element of X less than t, and t 1 denotes the smallest
element of X greater than t. Let N 1 , N 2 , N 3 denote the number of observations falling into
these three sets, respectively. Then

m m n+k−m

Pr(X (m) = t) = Pr(N 1 = m − k, N 2 ≥ k) = Pr(N 1 = m − k, N 2 = j).
k=1 k=1 j=k
Note that (N 1 , N 2 ) has a multinomial distribution with
n

n 2
n 3
n 1
Pr(N 1 = n 1 , N 2 = n 2 ) = F(t 0 ) p(t) (1 − F(t)) ,
n 1 , n 2 , n 3
n 1 + n 2 + n 3 = n, where F and p denote the distribution function and frequency function,
respectively, of the distribution of X 1 . Hence,
Pr(X (m) = t)
m n+k−m
n m−k j n−m+k− j
= F(t 0 ) p(t) (1 − F(t))
m − k, j, n − m + k − j
k=1 j=k
m n+k−m
m−k n n − m + k j n−m+k− j
= F(t 0 ) p(t) (1 − F(t)) ,
m − k j
k=1 j=k
the result in part (i).
Now suppose that X 1 has an absolutely continuous distribution. Recall from Theorem
7.6 that X (m) has distribution function

n
n
i n−i
F (m) (t) = F(t) (1 − F(t)) ,
i
i=m