Expected Value of a Function of a Random Variable
Definition
Let X be a discrete random variable with PMF $$\space\space P(X = x) = p_x, \quad x \in \mathbb{R} .$$ The expected value of a function of a random variable, $g(X)$, is defined as $$E[g(X)] = \sum_{x \in \mathbb{R}} g(x) P(X = x)$$
Properties
The expected value of a function of a random variable has the following properties: * Linearity: $E[a g(X) + b h(X)] = a E[g(X)] + b E[h(X)]$ for constants $a$ and $b$. * Non-negativity: $E[g(X)] \ge 0$ if $g(x) \ge 0$ for all $x$.
Applications
The expected value of a function of a random variable is used in a variety of applications, including: * Finding the mean and variance of a random variable. * Calculating probabilities. * Making decisions under uncertainty.
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