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2. Expectations of Functions of Random Variables 73

2.2.3 The Poisson Distribution
–λ x
Suppose that X has the Poisson (λ) distribution with its pmf f(x) = e λ /x!
where x = 0, 1, ..., 0 < λ < ∞, given by (1.7.4). In order to find the mean and
variance of this distribution, one may proceed in the same way as in the bino-
mial case. After the dust settles, one has to find the sums of the following two
infinite series:

2
Now, and similarly II = λ .
Here, the exponential series expansion from (1.6.15) helps. The details are left
out as the Exercise 2.2.7. Finally, one has

2.2.4 The Uniform Distribution

Suppose that X has the Uniform (α, β) distribution with its pdf f(x) = (β – α) –1
for α < x < β, given by (1.7.12). Here α, β are two real numbers. In order to
find the mean and variance of this distribution, we proceed as follows. Let us
write

The preceding answer should not be surprising. The uniform distribution puts
the same density or weight, uniformly on each point x ∈ (α, β), so that we
should expect the midpoint of the interval (α, β) to be designated as the mean or
the center of this distribution. Next, we write

Now we combine the Theorem 2.2.2 and (2.2.19)-(2.2.20) to claim that

Finally, we summarize:

2.2.5 The Normal Distribution
First, let us consider the standard normal random variable Z having its pdf
for –∞ < z < ∞, given by (1.7.16).