Page 229 - Elements of Distribution Theory

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

7.4 Sums of Random Variables 215

Since
n

n! n
x 1 ,...,x m
= = ,
s

s!x m ! s
x 1 ,...,x m−1
it follows that
x 1
x 2 x m
n s n−s s θ 1 θ 2 θ m
p S (s) = η (1 − η) ··· .
s x 1 ,..., x m−1 η η η
X s
Note that
x 1 x m−1
s θ 1 θ m−1
···
η η
x 1 ,..., x m−1
is the frequency function of a multinomial distribution with parameters s and
θ 1 /η, . ..,θ m−1 /η.
Hence,
x 1 x m−1
s θ 1 θ m−1
··· = 1
x 1 ,..., x m−1 η η
X s
and, therefore,
n s n−s

p S (s; θ 1 ,...,θ m ) = η (1 − η) , s = 0,..., n
s
so that S has a binomial distribution with parameters n and m−1 θ j .
j=1
Example 7.18 (Sum of uniform random variables). Let X 1 , X 2 denote independent, iden-
tically distributed random variables, each with a uniform distribution on the interval (0, 1);
hence, (X 1 , X 2 ) has an absolutely continuous distribution with density function
2
p(x 1 , x 2 ) = 1, (x 1 , x 2 ) ∈ (0, 1) .
Let S = X 1 + X 2 . Then S has an absolutely continuous distribution with density function
1
p S (s) = I {0<s−x 2 <1} I {0<x 2 <1} dx 2 .
0
Note that p S (s)is nonzero only for 0 < s < 2. Suppose 0 < s ≤ 1; then
s
p S (s) = dx 2 = s.
0
Suppose 1 < s < 2, then

1
p S (s) = dx 2 = 2 − s.
s−1
It follows that S has density function
0 if s ≤ 0or s ≥ 2

p S (s) = s if 0 < s ≤ 1 .
2 − s 1 < s < 2
The distribution of S is called a triangular distribution.