let x be a random variable that represent number of heads obtained in tossing n coins and (omega,F,P) be a probability space consider the convex function f(x)=x^2+2 defined on R Verify Jensens inequality
To verify Jensen's inequality for the convex function f(x) = x^2 + 2, we need to show that for any random variable X and for any function g(x), satisfying certain conditions, the following inequality holds:
E[f(X)] ≥ f(E[X])
Let's break down the steps to verify this inequality:
1. Start with the definition of expectation:
E[f(X)] = ∑[x∈R] f(x) * P(X=x)
2. Apply the definition of f(x) = x^2 + 2:
E[f(X)] = ∑[x∈R] (x^2 + 2) * P(X=x)
3. Use the linearity of expectation:
E[f(X)] = ∑[x∈R] x^2 * P(X=x) + 2 * ∑[x∈R] P(X=x)
4. Rewrite the second term using the probability sum rule:
E[f(X)] = ∑[x∈R] x^2 * P(X=x) + 2 * ∑[x∈R] P(X=x)
= ∑[x∈R] x^2 * P(X=x) + 2 * 1 (since ∑[x∈R] P(X=x) = 1)
5. Use the definition of expectation again:
E[f(X)] = E[X^2] + 2
6. Now, let's calculate E[X] to compare with f(E[X]):
E[X] = ∑[x∈R] x * P(X=x)
7. Apply Jensen's inequality for convex functions:
f(E[X]) = (E[X])^2 + 2
8. Finally, compare E[f(X)] with f(E[X]):
E[X^2] + 2 ≥ (E[X])^2 + 2
9. Simplify the inequality:
E[X^2] ≥ (E[X])^2
10. This inequality is the variance definition:
Var(X) = E[X^2] - (E[X])^2 ≥ 0
Therefore, Jensen's inequality holds for f(x) = x^2 + 2, and it is equivalent to the non-negativity of the variance.