Chapter 5: Calculus and Probability

1. Define the expectation of a function of a continuous random variable, and compute such expectations using elementary techniques from integral calculus (\(u\)-substitution, integration by parts, etc.).
2. Recognize the direct correspondence between differences and sums for discrete random variables and derivatives and integrals for continuous random variables.
3. State the ‘survival function trick’ for computing the expectation of a continuous random variable, and recognize problems where this trick is appropriate.
4. Define the mode of a continuous random variable, and determine the mode using techniques for optimizing functions of a single variable.
5. Recognize the shortcut to finding a mode on Exam P.
6. Define the \(100p\)-th percentile of a continuous random variable, and determine the \(100p\)-th percentile given either the random variable’s probability density function or cumulative distribution function.
7. Recognize the shortcut to finding a percentile on Exam P.
8. Distinguish between discrete, continuous, and mixed-type random variables in terms of the properties of their respective cumulative distribution functions.
9. Sketch the graph of a cumulative distribution function for a mixed-type random variable, and indicate where the mass for probability atoms lies in the graph.
10. State the basic form for a random variable defined as the mixture of two random variables, and state its cumulative distribution function and expected value.