1. Let X be uniform on the set {1/2,1/3,1/4}, and let the distribution of Y |X = x be geometric with parameter x; that is, P(Y = k|X = x) = (1 - x)kx, for k = 0,1,2,.... 
(a) What is the probability mass function of Y ? 
(b) What is the mean of Y ? 
(c) Find the probability mass function of X|Y = k for k = 0,1,2,... 
2. A company that manufactures cogs sells them in cartons of 100. It is historically known that about 1% of the cogs manufactured by the company are defective. 
(a) Write an exact expression for the probability that a carton has more than 2 defective cogs in it. 
(b) Approximate this same probability using the normal distribution. 
(c) Approximate this same probability using the Poisson distribution. 
(d) Which of the approximations of parts (b) and (c) is better? Why is this the case? 
3. Assume that Q is a random variable with density proportional to q for 0 < q < 1. Given Q = q, N has a binomial distribution with parameters n and q. The following identity for a,b > -1 is useful for this problem: 
. 
1 
(a) What is the probability mass function of N? 
(b) What is the mean of N and the expected value of 1/(N + 1) ? 
(c) Find the density of Q|N = k for k = 0,1,2,...,n 
4. The lifetime of a certain type of electrical component is well modeled by an exponential distribution with mean 2 hours. Assume that a system uses one component of this type and when there is a failure, the component is immediately replaced by another of the same type with exponential lifetime independent of all previous components. 
(a) Given the first component has not failed after two hours, what is the chance it will not fail within the following two hours? 
(b) What is the chance that at least two components fail in the first hour? 
5. Assume that X and Y are independent exponential random variables with rates 1 and 2, respectively. 
(a) Find the distribution function of the maximum of X and Y . 
(b) Find the density of the minimum of X and Y . Can you recognise the distribution of the minimum by name? 
(c) What is the probability that X < Y ? 
6. Let (X,Y ) be a point uniformly chosen on the unit disc {(x,y) : x2 + y2 < 1}. 
(a) Write down the density of (X,Y ). 
(b) What is the density of the angle 0 < T < 2p made between the positive x-axis and the ray connecting the origin to the point (X,Y ) ? 
(c) What is the density of X? 
(d) What is the density of Y |X = x for -1 < x < 1 ? 
2