Page 230 - Elements of Distribution Theory

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216 Distribution Theory for Functions of Random Variables

Example 7.19 (Dirichlet distribution). Let (X, Y) denote a two-dimensional random vec-
tor with an absolutely continuous distribution with density function
(α 1 + α 2 + α 3 ) α 1 −1 α 2 −1 α 3 −1
p(x, y) = x y (1 − x − y) ,
(α 1 ) (α 2 ) (α 3 )
where x > 0, y > 0, x + y < 1; here α 1 ,α 2 ,α 3 are positive constants. Let Z = 1 − X − Y;
then the distribution of (X, Y, Z)isanexample of a Dirichlet distribution.
Consider the distribution of X + Y. According to Theorem 7.5, S has an absolutely
continuous distribution with density function

(α 1 + α 2 + α 3 ) s α 1 −1 α 2 −1 α 3 −1
p S (s) = (s − y) y (1 − s) dy
(α 1 ) (α 2 ) (α 3 ) 0
(α 1 + α 2 + α 3 ) α 1 +α 2 −1 α 3 −1 1 α 1 −1 α 2 −1
= s (1 − s) (1 − u) u du
(α 1 ) (α 2 ) (α 3 ) 0
(α 1 + α 2 + α 3 ) α 1 +α 2 −1 α 3 −1 (α 1 ) (α 2 )
= s (1 − s)
(α 1 ) (α 2 ) (α 3 ) (α 1 + α 2 )
(α 1 + α 2 + α 3 ) α 1 +α 2 −1 α 3 −1
= s (1 − s) , 0 < s < 1.
(α 1 + α 2 ) (α 3 )
This is the density function of a beta distribution; see Exercises 4.2 and 5.6.

7.5 Order Statistics
Let X 1 ,..., X n denote independent, identically distributed, real-valued random variables.
The order statistics based on X 1 , X 2 ,..., X n , denoted by X (1) , X (2) ,..., X (n) , are simply
the random variables X 1 , X 2 ,..., X n placed in ascending order. That is, let denote the
underlying sample space of the experiment; then, for each ω ∈ ,

X (1) (ω) = min{X 1 (ω),..., X n (ω)},
X (2) (ω)is the second smallest value from X 1 (ω),..., X n (ω) and so on, up to X (n) (ω),
the maximum value from X 1 (ω),..., X n (ω). Hence, the random variables X (1) ,..., X (n)
satisfy the ordering

X (1) ≤ X (2) ≤ ··· ≤ X (n) .
There are at least two ways in which order statistics arise in statistics. One is that process
generating the observed data might involve the order statistics of some of underlying, but
unobserved, process. Another is that order statistics are often useful as summaries of a set
of data.

Example 7.20 (A model for software reliability). Consider the model for software reli-
ability considered in Example 6.15. In that model, it is assumed that a piece of software
has M errors or “bugs.” Let Z j denote the testing time required to discover bug j, j =
1,..., M. Assume that Z 1 , Z 2 ,..., Z M are independent, identically distributed random
variables.