Chapter 3: Discrete Random Variables

1. Define a random variable in terms of the probability theory we developed in Chapter 2, in particular noting that a random variable is a function from the sample space to the real line.
2. Correctly use the notation of uppercase Roman letters (\(X, Y, Z\)) for random variables, lowercase Roman letters (\(x, y, z\)) for values in the range of a random variable, and calligraphic Roman letters (\(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\)) for the range of a random variable.
3. State what requirement is placed on the range of a discrete random variable.
4. Define the probability mass function of a discrete random variable both in terms of an event from the underlying sample space and in how it is commonly used.
5. Specify the requirements on a function for it to correspond to the probability mass function of some discrete random variable.
6. Define the cumulative distribution function of a discrete random variable both in terms of an event from the underlying sample space and in how it is commonly used.
7. Specify the four main properties of a cumulative distribution function.
8. Given a probability mass function, compute the corresponding cumulative distribution function, and vice versa.
9. Define the expectation of a function of a discrete random variable.
10. Recognize how to decompose the expectation of a linear combination in terms of the linear combination of the expectations.
11. State the ‘survival function trick’ for computing an expectation, and recognize problems where this trick is appropriate.
12. Define the variance of a discrete random variable.
13. Compute the variance of a discrete random variable using the ‘variance shortcut.’
14. State how to decompose the variance of \(a X + b\) in terms of the variance of \(X\).
15. State how to standardize a random variable using its mean and standard deviation.
16. State definition of the coefficient of variation.
17. State Markov’s and Chebyshev’s inequalities.