Expected Values of Discrete Random Variables

Introduces the expected value (mean) of a discrete random variable as a probability-weighted average. Covers the basic formula $E[X]=\sum x_i p_i$ , the law-of-the-unconscious-statistician formula $E[g(X)]=\sum g(x_i)p_i$ , applications to games and net winnings, and recovering unknown parameters from a known expected value.

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Defining the Expected Value

The expected value (or mean) of a discrete random variable $X$ is the weighted average of its possible values, where each value is weighted by its probability.

If $X$ takes values $x_1, x_2, \ldots, x_n$ with corresponding probabilities $p_i = P(X=x_i),$ then

E[X] = \sum\limits_{i=1}^n x_i\, p_i = x_1 p_1 + x_2 p_2 + \cdots + x_n p_n.

For instance, suppose $X$ has the PMF

\begin{array}{c|ccc} x_i & -1 & 0 & 2 \\ \hline p_i & 0.5 & 0.2 & 0.3 \end{array}

Then

E[X] = (-1)(0.5) + 0(0.2) + 2(0.3) = -0.5 + 0 + 0.6 = 0.1.

The probabilities must sum to $1,$ and each $p_i \geq 0.$