Math 664 Homework #2: Solutions 1. Let Ω = R, F = {A ⊆ R : either A ...

4. Use Jensen’s inequality and size-biasing to show that if X ≥ 0 is a random variablesuch that0 < ρ = E[X] < ∞, σ 2 = V ar(X) < ∞, and σ ρ ≤ Mfor a given constant M > 0, thenρ 3/2 ≤ E[X 3/2 ] ≤ C · ρ 3/2for some constant C > 1, e.g. C = √ 1 + M 2 .SOLUTION: Since ( x 3/2) ′′and by Jensen’s inequality,= 34 √ x > 0 for x > 0, function x3/2 is convex on [0, ∞)ρ 3/2 ≤ E[X 3/2 ]For the upper bound, we will use size-biasing. Let∫xν(A) =ρ dµ(x) ∀A ∈ BThen ν is a probability measure over (R, B), and ν ≪ µ.ANext, we apply Jensen’s inequality for the concave function √ x and obtainE µ [X 3/2 ] = E µ [X · √X]= ρ · E ν [ √ X] ≤ ρ · √Eν [X] = ρ ·√E µ [X 2 ]ρ= ρ ·√σ 2 + ρ 2ρThereforeE µ [X 3/2 ] ≤ ρ 3/2 ·√σ 2 + ρ 2ρ 2 ≤ ρ 3/2 · √1+ M 2