Variance
This problem will give you practice using the "standard method" to compute the variance of a sum of random variables that are not pairwise independent. Recall that var(X) = E[X2]−E[X]2.
A building has n floors numbered 1,2,...,n, plus a ground floor G. At the ground floor, m people get on the elevator together, and each person gets off at one of the n floors uniformly at random (independently of everybody else). What is the variance of the number of floors the elevator does not stop at? (In fact, the variance of the number of floors the elevator does stop at must be the same, but the former is a little easier to compute.)
2 Probabilistically Buying Probability Books
Chuck will go shopping for probability books for K hours. Here, K is a random variable and is equally likely to be 1, 2, or 3. The number of books N that he buys is random and depends on how long he shops. We are told that
(kc for n = 1,...,k
P[N = n|K = k] =
0 otherwise
for some constant c.
(a) Compute c.
CS 70, Fall 2018, DIS 9B 1
(b) Find the joint distribution of K and N.
(c) Find the marginal distribution of N.
3 Correlation and Independence
(a) What does it mean for two random variables to be uncorrelated?
(b) What does it mean for two random variables to be independent?
(c) Are all uncorrelated variables independent? Are all independent variables uncorrelated? If your answer is yes, justify your answer; if your answer is no, give a counterexample.
