Page 99 - Probability and Statistical Inference

76 2. Expectations of Functions of Random Variables

2.2.6 The Laplace Distribution
Suppose that a random variable X has the Laplace or the double exponential
+
pdf f(x) = 1/2β exp{– |x – θ|/β} for all x ∈ ℜ where θ ∈ ℜ and β ∈ ℜ . This
pdf is symmetric about x = θ. In order to evaluate E(X), let us write

This reduces to θ since . Next, the variance, V(X)
is given by

One can see that and hence the last step of (2.2.32) can be rewritten as

with Γ(.) defined in (1.6.19). For the Laplace distribution, we then summarize
our findings as follows:

2.2.7 The Gamma Distribution
We consider a random variable X which has the Gamma (α, β) distribution
α
–1 –x/β α–1
with its pdf f(x) = {β Γ(α)} e x for 0 < x < ∞, given by (1.7.20). Here, we
+
+
have (α, β) ∈ ℜ × ℜ . In order to derive the mean and variance of this distri-
bution, we proceed with the one-to-one substitution u = x/β along the lines of
(1.7.22) where x > 0, and express E(X) as

In other words, the mean of the distribution simplifies to