The expected value or mean of a random variable $g(X),$ denoted by $\E(g(X)),$ is \[\E(g(X)) = \begin{cases} \int_{-\infty}^{\infty} g(x) f_X(x)\, dx & \text{if } X \text{ is continuous} \\ \sum_{x\in\mathcal{X}} g(x)f_X(x) = \sum_{x\in\mathcal{X}} g(x) P(X=x) & \text{if } X \text{ is discrete} \end{cases}\] provided that the integral or sum exists. If $\E|g(X)| = \infty,$ we say that $\E(g(X))$ does not exist.
Theorem 2.2.1
Let $X$ be a random variable and let $a,b,$ and $c$ be constants. Then for any functions $g_1(x)$ and $g_2(x)$ whose expectations exist,
a.) $\E(ag_1(X) + bg_2(X) + c) = a\E(g_1(X)) + b\E(g_2(X)) + c$
b.) If $g_1(x) \geq 0$ for all $x,$ then $\E(g_1(X))\geq 0.$
c.) If $g_1(x) \geq g_2(x)$ for all $x,$ then $\E(g_1(X)) \geq \E(g_2(X)).$
d.) If $a \leq g_1(x) \leq b$ for all $x,$ then $a \leq E(g_1(X)) \leq b.$
a.) $\E(ag_1(X) + bg_2(X) + c) = a\E(g_1(X)) + b\E(g_2(X)) + c$
b.) If $g_1(x) \geq 0$ for all $x,$ then $\E(g_1(X))\geq 0.$
c.) If $g_1(x) \geq g_2(x)$ for all $x,$ then $\E(g_1(X)) \geq \E(g_2(X)).$
d.) If $a \leq g_1(x) \leq b$ for all $x,$ then $a \leq E(g_1(X)) \leq b.$