Page 91 - Probability and Statistical Inference

68 2. Expectations of Functions of Random Variables

where χ is the support of X and µ = E(X). The variance is frequently assigned
the symbol σ . The positive square root of σ , namely σ, is called the standard
2
2
deviation of X.
The two quantities µ and σ play important roles in statistics. The variance
2
measures the average squared distance of a random variable X from the center
µ of its probability distribution. Loosely speaking, a large value of σ indicates
2
that on the average X has a good chance to stray away from its mean µ,
whereas a small value of σ indicates that on the average X has a good chance
2
to stay close to its mean µ.
Next, we state a more general definition followed by a simple result. Then,
2
we provide an alternative formula to evaluate σ .
Definition 2.2.3 Start with a random variable X and consider a function
g(X) of the random variable. Suppose that f(x) is the pmf or pdf of X where x
∈ χ. Then, the expected value of the random variable g(X), denoted by E(g(X)),
E{g(X)} or E[g(X)], is defined as:

where χ is the support of X.
Theorem 2.2.1 Let X be a random variable. Suppose that we also have
real valued functions g (x) and constants a , i = 0, 1, ..., k. Then, we have
i i

as long as all the expectations involved are finite. That is, the expectation is a
linear operation.
Proof We supply a proof assuming that X is a continuous random variable
with its pdf f(x), x ∈ χ. In the discrete case, the proof is similar. Let us write

constant and using property of the integral operations.
Hence, we have