Prove that 𝐄[X^k] ≥𝐄[X]^k for any even integer k ≥1 |...

Name: Problem 4, Chapter 2: Probability and Computing
Uploaded: 2020-03-30T17:26:34-08:00
Duration: 1 min 58 s
Channel: Julian Wong
Description: Prove that 𝐄[X^k] ≥𝐄[X]^k for any even integer k ≥1

Key Concepts

Jensen's Inequality

Jensen's inequality states that for any convex function f and any random variable X, the inequality f(E[X]) ? E[f(X)] holds true. This principle is crucial when working with expectations, particularly because it allows us to compare the application of a function to an expectation with the expectation of that function applied to a random variable. In this problem, using Jensen's inequality helps to establish that the expected value of X raised to an even power is at least as large as the power of the expected value of X.

Convex Functions

A function is considered convex if the line segment between any two points on its graph lies above or on the graph. Mathematically, a function f is convex if for all x and y in its domain and for all t in [0, 1], the relation f(tx + (1-t)y) ? t f(x) + (1-t) f(y) holds. For even integers k, the function x^k is convex over all real numbers, which is a key fact in applying Jensen's inequality to prove the desired inequality in the question.

Moments of a Random Variable

The k-th moment of a random variable X is defined as E[X^k] and provides important information about the distribution's shape, including aspects like its spread and tails. In the context of this problem, comparing the k-th moment to (E[X])^k offers insight into how the dispersion of the random variable affects its higher order behaviors. The inequality E[X^k] ? (E[X])^k for even k demonstrates a fundamental relationship in moment theory underpinned by the convexity of the power function.

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