Homework I, Undergraduate Course Review Problems

Problems 1-4 are due Friday, October 1, the rest on Thursday, October 7.

Problem 1:

Suppose a die is rolled n times, resulting in X 3s. Then it is rolled m more times, resulting in Y evens (2, 4 or 6). Let Z = X+Y.

  1. Consider the case n = 3, m = 2. Find pX,Z(2,3). Give reasons, by citing equation numbers as in Equations (2.7)-(2.10). (You need not do this in other problems, just in this warmup.) Check your answer using R simulation.

  2. Now assume general n and m. Find a closed-form expressions for pX,Z and Var(Z). Verify your expression for the latter via R simulation, for some (n,m) pair of your choice.

Problem 2:

  1. Exercise 14, Chapter 3.

  2. Use (a) to prove that if Y &ge X with probability 1 for discrete nonnegative random variables X and Y, then EY &ge EX. (The relation is true even if X or Y is not nonnegative, and can be seen easily on an intuitive level via a "notebook argument.")

Problem 3:

Exercises 6 and 10, Chapter 3.

Problem 4:

Suppose a bit stream is read by three receivers, acting independently. For any bit and any receiver, there is a probability p that the bit will be received in error. We pool the three received versions of a bit by using "majority rule": If two or three of the receivers say the bit is a 1, we decide it is a 1, and similarly for a 0. Assume independence from one bit to the next. Find the distribution of W, the number of errors that this "majority rule" will produce in an n-bit stream.

Problem 5:

Exercise 12, Chapter 4.

Problem 6:

Suppose X has an exponential distribution with mean 1.0, and let N = ceil(X/c), for fixed c > 0. Show that the distribution of N belongs to a famous parametric family.

Problem 7:

Suppose X and Y are independent, with each having density 2t on the interval (0,1), 0 elsewhere. We are interested in the distribution of (X,Y) given the event A = {X+Y > 1}.

  1. Find the conditional density of (X,Y), given A, i.e. fX,Y|A(u,v).

  2. Find Var(X) and Var(X|A). Comment.

  3. Find ρ(X,Y) and ρ(X,Y|A). Comment.

  4. Write R simulation code to confirm your answers in (b) and (c).