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Chapter 2: Problem 48

Prove that \(E\left[X^{2}\right] \geq(E[X])^{2}\). When do we have equality?

Short Answer

Expert verified
Using Jensen's Inequality on the convex function \(f(x) = x^2\), we obtain the inequality \(E[X^2] \geq (E[X])^2\). The equality holds if and only if X is a constant random variable.

Step by step solution

01

Identify the convex function

We will use the convex function f(x) = x^2. A function is convex if it satisfies the following condition for any two points x, y and any t 鈭� (0, 1): f(tx + (1-t)y) 鈮� tf(x) + (1-t)f(y) In our case, f(x) is a quadratic function with a positive leading coefficient, which makes it a convex function.
02

Apply Jensen's Inequality

Now, let's apply Jensen's Inequality to the function f(x) and the random variable X. Since f(x) is convex, we have: \(E[f(X)] \geq f(E[X])\) Substituting f(x) = x^2, we get: \(E[X^2] \geq (E[X])^2\)
03

Determine when equality holds

The equality in Jensen's Inequality holds if and only if X is a constant random variable or if the function f is linear on the support of the random variable X. In our case, the function f(x) = x^2 is not linear. Therefore, the equality holds if and only if X is a constant random variable. So the inequality \(E[X^2] \geq (E[X])^2\) holds for any random variable X, and we have equality if and only if X is a constant random variable.

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