Page 175 - Probability and Statistical Inference

152 3. Multivariate Random Variables

3.9.4 Jensen’s and Lyapunov’s Inequalities
Consider a real valued random variable X with finite variance. It is clear that

2
2
which would imply that E[X ] ≥ E [X]. That is, if we let f(x) = x , then we can
2
claim

The question is this: Can we think of a large class of functions f(.) for which
the inequality in (3.9.16) would hold? In order to move in that direction, we
start by looking at the rich class of convex functions f(x) of a single real
variable x.
Definition 3.9.1 Consider a function f: ℜ → ℜ. The function f is called
convex if and only if

for all u, v ∈ ℜ and 0 ≤ α ≤ 1. The function f is called concave if and only if
– f is convex.

Figure 3.9.1. A Convex Function f(x)

The situation has been depicted in the Figure 3.9.1. The definition of the
convexity of a function demands the following geometric property: Start
with any two arbitrary points u, υ on the real line and consider the chord
formed by joining the two points (u, f(u)) and (υ, f(υ)). Then, the function
f evaluated at any intermediate point z, such as αu + (1 – α)υ with 0 ≤