Page 227 - Elements of Distribution Theory

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

7.4 Sums of Random Variables 213

It follows that the characteristic function of X = (X 1 ,..., X n )is
n

ϕ X (t 1 ,..., t n ) = ϕ(t i ).
j=1
n
Hence, the characteristic function of S = X j is given by
j=1
λ n
n
ϕ S (t) = ϕ(t) = .
(λ − it) n
This is the characteristic function of the gamma distribution with parameters n and λ; see
Example 3.4. It follows that S has a gamma distribution with parameters n and λ.

The techniques described in the previous section can be used to find the density or
frequency function of a sum.

Theorem 7.5. Let X = (X 1 ,..., X n ) where X 1 ,..., X n denotes a sequence of real-valued
n

random variables, let S = X j and let S denote the range of S.
j=1
(i) Suppose X has a discrete distribution with frequency function p. Then S has a
discrete distribution with frequency function p S where

p S (s) = p(x 1 ,..., x n ), s ∈ S
n
{(x 1 ,...,x n ): j=1 x j =s}
(ii) Suppose X has an absolutely continuous distribution with density function p. Then
S has an absolutely continuous distribution with density function p S where

∞ ∞
n

p S (s) = ··· p s − x j , x 2 ,..., x n dx 2 ··· dx n , s ∈ R
−∞ −∞ j=2
Proof. Let f denote a bounded, real-valued function defined on S, the range of S. Then,
if X has discrete distribution with frequency function f ,

E[ f (S)] = f (s)p(x 1 ,..., x n ) = f (s)p(x 1 ,..., x n )
n
(x 1 ,...,x n )∈X s∈S {(x 1 ,...,x n ):
j=1 x j =s}

= f (s) p(x 1 ,..., x n );
n
s∈S {(x 1 ,...,x n ): j=1 x j =s}
part (i) of the theorem follows.
Now suppose that X has an absolutely continuous distribution with density function p.
To prove part (ii), we can use Theorem 7.2 with the function
n

g(x) = (s, x 2 ,..., x n ), s = x j .
j=1
Then Y = g(X) has density
p(y 1 − (y 2 +· · · + y n ), y 2 ,..., y n );

note that the Jacobian here is equal to 1. Hence, the marginal density of Y 1 = S is
∞ ∞

··· p(y 1 − (y 2 +· · · + y n )y 2 ,..., y n ) dy 2 ··· dy n ;
−∞ −∞
rewriting this in terms of s = y 1 and x j = y j , j = 2,..., n, proves the result.