Chapter 3: Discrete Random Variables

1. Construct the probability mass function / cumulative distribution function for a discrete random variable conditional on it belonging to some set.
2. Explain why we need to renormalize the probability mass function / cumulative distribution function when we condition on the random variable belonging to some set.

Review of Finite Sums and Series

1. Define an arithmetic sequence, and identify an arithmetic sequence showing up in a partial sum.
2. Recall the ‘trick’ to computing the partial sum of an arithmetic sequence.
3. State the closed-form expression for the partial sum of an arithmetic sequence.
4. Define a geometric sequence, and identify a geometric sequence showing up in a partial sum.
5. Recall the ‘trick’ to computing the partial sum of a geometric sequence.
6. State the closed-form expression for the partial sum of a geometric sequence.
7. Recall the power series expansions of some common functions, such as \((x + a)^{n}, e^{x}, \frac{1}{1 -x}.\)
8. Derive the power series expansion of a function \(f\) at a point \(x_{0}\) using the explicit formula for a Taylor series.
9. Manipulate the known power series expansion for a function \(f\) to derive the power series expansion for related functions.
10. Use the derivative and antiderivative of the power series expansion of a function to determine the power series expansion of its derivative and antiderivative.