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Definition 1. A function $g(x)$ is called convex if for $x, y in mathbb{R}$ and $t in [0, 1]$, $g(tx + (1 - t)y) le tg(x) + (1 - t)g(y)$. Fact 2. If $g(x)$ is twice differentiable, then $g(x)$ is convex if and only if for all $x$, $g''(x) ge 0$. Theorem 3 (Jensen's inequality). Suppose $g$ is a convex function. Let $X$ be a RV with $E|X| < infty$, then $g(E[X]) le E[g(X)]$. Example: $g(x) = x^2$ is convex (since $g''(x) = 2 > 0$). Thus, $g(E[X]) = (E[X])^2 le E[g(X)] = E[X^2]$. 1. (a) Verify that $x^2$ is a convex function by directly using Definition 1. (b) Without using convexity, prove that $(E[X])^2 le E[X^2]$.

          Definition 1. A function $g(x)$ is called convex if for $x, y in mathbb{R}$ and $t in [0, 1]$,
$g(tx + (1 - t)y) le tg(x) + (1 - t)g(y)$.
Fact 2. If $g(x)$ is twice differentiable, then $g(x)$ is convex if and only if for all $x$, $g''(x) ge 0$.
Theorem 3 (Jensen's inequality). Suppose $g$ is a convex function. Let $X$ be a RV with $E|X| < infty$, then
$g(E[X]) le E[g(X)]$.
Example: $g(x) = x^2$ is convex (since $g''(x) = 2 > 0$). Thus,
$g(E[X]) = (E[X])^2 le E[g(X)] = E[X^2]$.
1. (a) Verify that $x^2$ is a convex function by directly using Definition 1.
(b) Without using convexity, prove that
$(E[X])^2 le E[X^2]$.

Definition 1. A function g(x) is called convex if for x, y in mathbbR and t in [0, 1],
g(tx + (1 - t)y) le tg(x) + (1 - t)g(y).
Fact 2. If g(x) is twice differentiable, then g(x) is convex if and only if for all x, g”(x) ge 0.
Theorem 3 (Jensen's inequality). Suppose g is a convex function. Let X be a RV with E|X| < infty, then
g(E[X]) le E[g(X)].
Example: g(x) = x^2 is convex (since g”(x) = 2 > 0). Thus,
g(E[X]) = (E[X])^2 le E[g(X)] = E[X^2].
1. (a) Verify that x^2 is a convex function by directly using Definition 1.
(b) Without using convexity, prove that
(E[X])^2 le E[X^2].

Added by Patricia S.

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