Problem 1. Let X₀;X₁;X₂. . . be a Markov chain with state space f1; 2g and transition probabilities given as follows:

p₁₁ = 0:3; p₁₂ = 0:7; p₂₁ = 0:5; p₂₂ = 0:5:

Find the invariant probability π of the chain. Find P(X₂ = 1;X₃ = 2 j X₀ = 1).

Problem 2. Let X be a random variable such that

M(s) = a + be^2s + ce^4s; E[X] = 3; var(X) = 2:

Find a, b and c and the PMF of X.

Problem 3. Let X and Y be independent exponential random variables with a common parameter λ. Find the moment generating function associated with aX + Y , where a is a constant.

Problem 4. Let Y be exponentially distributed with parameter 1, and let Z be uniformly distributed over the interval [0; 1]. Find the PDFs of W = Y - Z and that of X = [Y - Z].

Problem 5. You roll a fair six-sided die, and then you flip a fair coin the number of times shown by the die. Find the expected value of the number of heads obtained.

Problem 6. The random variables X1. . . ;Xn have common mean μ, common variance σ² and, furthermore, E[XiXj ] = c for every pair of distinct i and j. Derive a formula for the variance of X1 + . . . + Xn, in terms of μ, σ², c, and n.

Problem 7. Consider n independent tosses of a die. Each toss has probability pi of resulting in i. Let Xi be the number of tosses that result in i. Find the covariance of X₁ and X₂.

Problem 8. We are given that E[X] = 1, E[Y ] = 2, E[X²] = 5, E[Y ²] = 8, and E[XY ] = 1. Find the linear least squares estimator of Y given X.

Problem 9. Let U and V be independent standard normal random variables, and X = U + V, Y = U - 2V. Find E[X | Y ], and cov(X, Y ).

Problem 10. A security guard has the only key which locks or unlocks the door to ENS. He visits the door once each hour on the hour. When he arrives: If the door is open, he locks it with probability 0.3. If the door is locked, he unlocks it with probability 0.8. After he has been on the job several months, is he more likely to lock the door or to unlock it on a randomly selected visit? With the process in the steady state, Joe arrived at Building 59 two hours ahead of Harry. What is the probability that each of them found the door in the same condition? Given the door was open at the time the security guard was hired, determine the expected value of the number of visits up to and including the one on which he unlocks the door himself for the first time.