Page 42 - Elements of Distribution Theory

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28 Properties of Probability Distributions

1.8 Expectation

Let X denote a real-valued random variable with distribution function F. The expected
value of X, denoted by E(X), is given by
∞

E(X) = xdF(x),
−∞
provided that the integral exists. It follows from the discussion in the previous section that,
if X has a discrete distribution, taking the values x 1 , x 2 ,... with frequency function p, then

E(X) = x j p(x j ).
j
If X has an absolutely continuous distribution with density p, then
∞

E(X) = xp(x) dx.
−∞
There are three possibilities for an expected value E(X): E(X) < ∞,E(X) =±∞,or
E(X) might not exist. In general, E(X)fails to exist if the integral
∞

xdF(x)
−∞
fails to exist; see Appendix 1. Hence, the expressions for E(X)given above are valid only
if the corresponding sum or integral exists. If X is nonnegative, then E(X)always exists,
although we may have E(X) =∞;in general, E(X)exists and is finite provided that
∞

E(|X|) = |x| dF(x) < ∞.
−∞
Example 1.27 (Binomial distribution). Let X denote a random variable with a binomial
distribution with parameters n and θ,as described in Example 1.4. Then X is a discrete
random variable with frequency function
n x n−x

p(x) = θ (1 − θ) , x = 0,..., n
x
so that
n
n x n−x
E(X) = x θ (1 − θ) = nθ.
x
x=0
Example 1.28 (Pareto distribution). Let X denote a real-valued random variable with an
absolutely continuous distribution with density function

p(x) = θx −(θ+1) , x ≥ 1,
where θ is a positive constant; this is called a Pareto distribution with parameter θ. Then
∞ ∞

E(X) = xθx −(θ+1) dx = θ x −θ dx.
0 0
Hence, if θ ≤ 1, E(X) =∞;if θ> 1, then
θ
E(X) = .
θ − 1