Page 208 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 185

4.2.4 The Convolution
We start with what is also called the convolution theorem. This result will
sometimes help to derive the distribution of sums of independent random
2
variables. In this statement, we assume that the support of (X , X ) is ℜ .
2
1
When the support is a proper subset of ℜ , the basic result would still hold
2
except that the range of the integral on the rhs of (4.2.9) should be appropri-
ately adjusted on a case by case basis.
Theorem 4.2.1 (Convolution Theorem) Suppose that X and X are in-
1
2
dependent continuous random variables with the respective pdf’s f (x ) and
1
1
f (x ), for (x , x ) ∈ ℜ . Let us denote U = X + X . Then, the pdf of U is given
2
1
1
2
2
2
2
by
for u ∈ ℜ.
Proof First let us obtain the df of the random variable U. With u ∈ ℜ, we
write

Thus, by differentiating the df G(u) from (4.2.10) with respect to u, we get
the pdf of U and write

which proves the result. ¢
If X and X are independent continuous random variables,
1
2
then the pdf of U given by (4.2.9) can be equivalently
written as instead.