Page 176 - Probability and Statistical Inference

3. Multivariate Random Variables 153

α ≤ 1, will lie under the chord.

Figure 3.9.2. A Convex Function f(x) and Tangent Lines (a), (b)
We may also look at a convex function in a slightly different way. A func-
tion f is convex if and only if the curve y = f(x) never goes below the tangent
of the curve drawn at the point (z, f(z)), for any z ∈ ℜ. This has been depicted
in the Figure 3.9.2 where we have drawn two tangents labelled as (a) and (b).
In the Figure 3.9.3, we have shown two curves, y = f(x) and y = g(x), x ∈ ℜ.
The function f(x) is convex because it lies above its tangents at any point
whatsoever. But, the function g(x) is certainly not convex because a

Figure 3.9.3. Two Functions: f(x) Is Convex, g(x) Is Not Convex

part of the curve y = g(x) lies below the chord as shown which is a violation
of the requirement set out in (3.9.17). In the Figure 3.9.3, if we drew the
tangent to the curve y = g(x) at the point (u, g(u)), then a part of the curve will
certainly lie below this tangent!
Example 3.9.8 Consider the following functions: (i) f(x) = x for x ∈ ℜ; (ii)
2
f(x) = e for x ∈ ℜ; (iii) f(x) = max(0, x) for x ∈ ℜ; and (iv) f(x) = log(x) for
x
x ∈ ℜ . The reader should verify geometrically or otherwise that the first
+
three functions are convex, while the fourth one is concave. !