Chapter 4: Some Discrete Distributions

1. State the probability mass function of a hypergeometric random variable \(X\) with parameters \((N, K, n)\), and state its mean and variance.
2. Relate the mean and variance of a hypergeometric random variable to the mean and variance of a binomial random variable, and explain how a hypergeometric random variable is ‘almost’ like a binomial random variable.
3. Identify when a hypergeometric or binomial random variable is most appropriate to solve a “\(x\) successes out of \(n\) trials” problem.
4. State what types of phenomena are well-modeled by Poisson random variables.
5. State the probability mass function of a Poisson random variable \(X\) with parameter \(\lambda\), and state its mean and variance.
6. State how sums of independent Poisson random variables behave.

Chapter 5: Calculus and Probability

1. State the necessary conditions for a random variable to be of continuous type.
2. Define the probability density function \(f\) for a continuous random variable.
3. State the sufficient conditions for a function \(f\) to be a probability density function for some continuous random variable, and check these conditions when given a candidate \(f\).
4. Specify how to compute \(P(X \in Q)\) for a continuous random variable \(X\) with probability density function \(f\) and a given query set \(Q\).
5. Define the cumulative distribution function \(F\) for a continuous random variable.
6. State the additional property that a cumulative distribution function for a continuous random variable has, beyond the four general conditions.
7. Derive \(F\) from \(f\) and vice versa.