1. X and Y arc two discrete random variables, with the P(X = j) for any non-negative integers i and j. where a and b are two positive parameters. (t) What is the marginal distribution of X? (ü) Vhat is the conditional distribution of Y given X .? (in) Derive Cov(X. Y).
2. Local and express trains arrive at a subway station following independent Poisson processes with rates A and p, respectively. Suppose each loca] train brings independently X passengers who want to transfer to the express train; suppose m1 := E(X) and m2 := E(X2) are known. Derive the mean and the variance of the number of passengers who will board the next express train (assuming unlimited capacity of the express train).
3. Consider the shuffling of a deck of 3 cards as follows;
(a) move the top card to the bottom, with probability p
(b) interchange the top and the middle cards, with probability 1 — p.
That is, cadi shuffle independently does either (a) or (b). with probability p awl 1 — p, respectively. The question is, after many such operations, whether the deck will be thoroughly shuffled. i.e., any order of the threc cards will be equally likely. To answer this question, construct a Markov chain with the state space being the 3?=6 different orderings of the cards.
(1) Write down the probability transition matrix, derive the limiting probabilities, and supply an answer to the above question.
(ii) Is there any restriction that we need to impose on p? For instance, do we need p = 0.5? Do we need p and/or p 1? Consider in particular: (a) when p 0, Is the Markov chain irreducible, and if not, what are the communicating classes; (b) repeat (a) for p 1.
4. A system has two servers with independent exponential service times with means 0.5 hour (server 1) and I hour (server 2). Jobs arrive at the vstem following a Poisson process with rate 2 per hour. A job will be served by server 1 if available; otherwise, it will be served by server 2. When both servers are busy, arrivals will be blocked and lost.
(1) Derive the average processing time of jobs that arc admitted Into the system.
(ii) Derive the average hourly output from each of the two servers, and verify that the total output rate is equal
to the rate of jobs admitted into the system
5. Denote X(t) (B(e) I B(s) = a, B(u) = bi for t E fs,uJ. where 0 < s < u. and a.b are given. What is the istribution olX(t), for any given t € [s,uJ? What Is E(X(t)j and VarfX(t)j?
6. An insurance company receives policy payments al rate p and pays out claims at rate c. Let p :— p — c and assume p > 0. Because of randomness involved, let X, := io ÷ pt + cli, denote the company’s cash position at time t. where B, is Brownian motion, o’> ()and zo > 0 are given constants, and z0 is the company’s initial cash posit ion.
(I) Let r be the first time X, drops down to zero, at which point the company Will default. That is the probability
for this to happen? What is E(r)?
(il) Suppose the company sets a growth target level of 2z0 + gt for Its cash position, where g € (0, p) Is a given constant. Let T be the first time the company meets the growth target. That is the probability for this to happen?
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