Section 3.3: Expected Values of Discrete Random Variables

1. Define the expected value of a discrete random variable \(X\), and compute \(X\)’s expected value given its probability mass function.
2. Give a frequency interpretation of the expected value of a random variable \(X\).
3. Recognize the convention of using Greek letters for parameters of a random variable, and their Roman counterpart for statistics of a sample. For example, \(\sigma\) for the standard deviation of a random variable and and \(s\) for the standard deviation of a sample.
4. Compute the expected value of a random variable \(Y = g(X)\) defined through a function \(g\) of a random variable \(X\) with a known probability mass function.
5. Simplify expectations of the form \(E[a X + b]\) and
   \(E\left[\sum\limits_{j = 1}^{n} g_{j}(X)\right]\) directly, without resorting to the definition of expectation.
6. Define the variance of a discrete random variable \(X\), and compute \(X\)’s variance given its probability mass function.
7. Define the standard deviation of a random variable.
8. Use the ‘variance shortcut’ to compute the variance of a random variable.
9. Simplify variances (standard deviations) of the form \(\text{Var}(a X + b)\) (\(\sigma_{a X + b}\)) directly, without resorting to the definition of variance (standard deviation).