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diff --git a/zhilova/08_jensen b/zhilova/08_jensen new file mode 100644 index 0000000..20a8158 --- /dev/null +++ b/zhilova/08_jensen @@ -0,0 +1,23 @@ +Definition: A fn. f : R -> R is called convex on an interval (a,b) if +f(cx + dy) \leq cf(x) + df(y) +\forall x, y \in (a, b) +\forall c \in (0, 1), d = 1-c. +Concave is -convex. + +Essentially stating that the function lies on or below a line segment +connecting f(a) and f(b) [or above in the case of concave]. + +Strictly convex: f(cx+dy) < cf(x) + df(y). + + Jensen's Inequality +X - r.v., E|X| < infty. E|f(x)| < infty. + f(E X) \leq E(f(x)). +If f is strictly convex, \leq -> "less than" unless X is a constant r.v. + +Further theorems: +(1) If f is differentiable on (a,b), +f is convex <=> f' is nondecreasing on (a,b). +f is strictly convex <=> f' is strictly increasing on (a,b) +(2) If f is twice differentiable on (a,b) +f is convex <=> f'' \geq 0 on (a,b) +f is strictly convex <=> f'' > 0 on (a,b) |