Chapter 7: Multivariate Distributions

1. State the definition of the conditional probability mass / density functions of a random vector \((X, Y)\).
2. Compute conditional probability mass / density functions given the the joint probability mass / density function of a random vector \((X, Y)\).
3. State, in words, the type of question answered by probability mass / density functions.
4. Given a scatter plot of \((x, y)\) points, roughly indicate the shape of the conditional probability density functions for the random vector \((X, Y)\) that might model those points.
5. Explain how a joint probability mass / density function for a random vector \((X, Y)\) can be “factored” in terms of its marginal and conditional probability mass / density functions.
6. Give the equivalent conditions for the independence of two random variables \(X\) and \(Y\).
7. State the necessary condition on the range of a random vector \((X, Y)\) for the random variables \(X\) and \(Y\) to be independent.
8. State the definition of mutual independence for \(n\) random variables \(X_{1}, X_{2}, \ldots, X_{n}\).
9. Explain why we can think of \(X \mid Y = y\) and \(Y \mid X = x\) as random variables.
10. Define and compute the conditional expectations of \(X\) given \(Y = y\) and \(Y\) given \(X = x\) for both discrete and continuous random vectors.
11. Define and compute the conditional variance of \(X\) given \(Y = y\) and \(Y\) given \(X = x\) for both discrete and continuous random vectors.
12. Given a scatter plot of \((x, y)\) points, sketch the functions \(E[Y \mid X = x]\) and \(E[X \mid Y = y]\).
13. Given a scatter plot of \((x, y)\) points, sketch the functions \(\text{Var}(Y \mid X = x)\) and \(\text{Var}(X \mid Y = y)\).