Page 238 - Elements of Distribution Theory

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

224 Distribution Theory for Functions of Random Variables

Let

Y 1 = X (1) , Y 2 = X (2) − X (1) ,..., Y n = X (n) − X (n−1) ;

note that, if X 1 ,..., X n denote event times of some random process, then X (1) ,..., X (n)
denote the ordered event times and Y 1 ,..., Y n denote the inter-event times.
The density function of Y = (Y 1 ,..., Y n ) can be obtained from Theorem 7.2. We can
write (X (1) ,..., X (n) ) = h(Y) ≡ (h 1 (Y),..., h n (Y)) where
h j (y) = y 1 + ··· + y j , j = 1,..., n.
It follows that the Jacobian of h is 1 and the density of Y is given by
n
p Y (y) = n!λ exp{−λ[y 1 + (y 1 + y 2 ) + ··· + (y 1 + y n )]}

n
n

= n!λ exp −λ (n − j + 1)y j , y j > 0, j = 1,..., n.
j=1
n
Since n! = (n − j + 1), it follows that Y 1 ,..., Y n are independent exponential ran-
j=1
dom variables such that Y j has parameter (n − j + 1)λ.
Example 7.26 (Range of uniform random variables). Let X 1 , X 2 ,..., X n denote inde-
pendent, identically distributed random variables, each distributed according to a uniform
distribution on (0, 1). Consider the problem of determining the distribution of X (n) − X (1) ,
the difference between the maximum and minimum values in the sample.
The joint distribution of (X (1) , X (n) )is absolutely continuous with density function

n(n − 1)(x n − x 1 ) n−2 , 0 < x 1 < x n < 1.
Let T = X (n) − X (1) and Y = X (1) . Then, using Theorem 7.2, the distribution of (T, Y)is
absolutely continuous with density function
p(t, y) = n(n − 1)t n−2 , 0 < y < t + y < 1.

Hence, the marginal density of T is
1−t
p T (t) = n(n − 1)t n−2 dy = n(n − 1)t n−2 (1 − t), 0 < t < 1.
0
Thus, the distribution of T is a beta distribution with parameters n − 1 and 2; see Example
7.24.

7.6 Ranks
Let X 1 ,..., X n denote independent, identically distributed, real-valued random variables
and let X (1) ,..., X (n) denote the corresponding order statistics. It is easy to see that
(X 1 ,..., X n ) and (X (1) ,..., X (n) ) are not equivalent statistics; in particular, it is not possible
to reconstruct X 1 ,..., X n given only X (1) ,..., X (n) . The missing information is the vector
of ranks of X 1 ,..., X n .
The rank of X i among X 1 ,..., X n is its position in the order statistics and is defined to
be the integer R i ,1 ≤ R i ≤ n, satisfying X i = X (R i ) , provided that X 1 ,..., X n are unique.
Here we assume that common distribution of X 1 , X 2 ,..., X n is absolutely continuous so