Section 4.1: Probability Density Functions and Cumulative Distribution Functions

1. Give examples of quantities that could be modeled using a continuous random variable.
2. Recognize and explain the correspondence between density histograms for data and probability density functions for random variables.
3. Given a probability density function \(f\) for a continuous random variable \(X\) and a query region \([a, b]\), determine \(P(X \in [a, b]) = P(a \leq X \leq b)\) using \(f\).
4. State the two sufficient conditions for a function \(f\) to be a valid probability density function, and identify when a given function \(f\) does or does not satisfy these conditions.
5. Given that a probability density function is proportional to a known function, i.e. \(f(x) = c g(x)\), determine the constant \(c\) that makes \(f\) a probability density function.
6. Define the cumulative distribution function of a continuous random variable, and compute the cumulative distribution function using the random variable’s probability density function.
7. Specify what additional property, beyond the four properties shared by all cumulative distribution functions, unique to the cumulative distribution function of a continuous random variable.
8. Compute a probability query \(P(X \in (a, b)) = P(a < X < b)\) using the cumulative distribution function of a continuous random variable.
9. Determine the probability density function of a continuous random variable from its cumulative distribution function.

Section 4.2: Expected Values and Moment Generating Functions

1. Define the expected value of a continuous random variable \(X\), and compute \(X\)’s expected value given its probability density function.
2. Recognize the correspondence between sums and probability mass functions for discrete random variables and integrals and probability density functions for continuous random variables.
3. Compute the expected value of a random variable \(Y = g(X)\) defined through a function \(g\) of a continuous random variable \(X\) with a known probability density function.
4. 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.
5. Define the variance of a continuous random variable \(X\), and compute \(X\)’s variance given its probability density function.
6. Use the ‘variance shortcut’ to compute the variance of a random variable.
7. 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).