1. A fair die is rolled until the first even outcome, (2, 4, 6), is obtained. Let Nequal the number of rolls required. For example, if we observe (1, 3, 3, 1, 5, 1, 5, 3, 6, . . .),then N = 9. Find

(a)  E (N|SN = 9)

(b) E(0.8)N

(c)  E (SN |N)

(d)  E(SN )

(e)  Var(SN )

2. Consider {Xi , i ≥ i} iid Exponential(2), {Yi , i ≥ i} iid Exponential(3), {N1(t), t ≥ 0} a Poisson process with intensity 1, and {N2(t), t ≥ 0} a Poisson process with intensity 2. Assume that {Xi}, {Yi}, {N1(t), t ≥ 0} and {N2(t), t ≥ 0} are independent. Define

(a) Is {W(t), t ≥ 0} a compound Poisson Process? Why or why not?

(b) Find P (W1(3) > W2(3)|N1(3) = 2, N2(3) = 3)

(c) Define 

(d) Define T3 to be the time of the 3rd event for the process {N(t), t ≥ 0} defined above. Find P (T3 > 1.5)

(e) Let N ∼ Bin(10, 0.3) with N independent of {Yi}. Find, 

3. Consider a standard Brownian motion process {X(t), t ≥ 0}. Find

(a) P (X(1) < 1 + X(3)).

(b) 

(c) Let Y denote the first zero for the Brownian motion process after time t = 5. Find P (Y > 8).

(d) Define X˜(t) = 3X(t) + 2t. Let T denote the first time that X˜(t) = √ 2, thus T = {inf t : X˜(t) = √ 2}. Find the mean and variance of T. 

(e) Find

4. Consider a Markov chain with state space {0, 1, 2, 3} and transition matrix

(a) Find lim k→∞ E [(X(k))X(k) |X(0) = 2]

(b) Define A = {1, 3}. Find the expected number of steps to go from 2 to A for the first time, that is find, 

(c) Find P (visit 0 before 3|X(0) = 2)

(d) Find P 4 (0, 0) = P (X(4) = 0|X(0) = 0)

(e) Find limn→∞ P (Xn+1 = 1, X2n = 2|X(0) = 3)

5.

(a)  Consider a continuous time Markov chain with state space {0, 1, 2} and infinitesimal matrix

Find an explicit expression for

P(t)01 = P (X(t) = 1|X(0) = 0)

(b) A factory has 2 machines and 2 repairmen. The operating time of a machine is exponential with mean 1 hour. The repair time of a machine is exponential with mean 15 minutes. A busy period begins when 1 of 2 working machines fail and ends the next time instant that both are working. Find the expected length of a busy period.

(c) We have 6 balls distributed among 2 urns, A and B. At each stage one of the urns is randomly chosen. If the chosen urn is non-empty then one of its balls is moved to the other urn. Otherwise, we do nothing. Find the long term proportion of time that one of the urns is empty.

(d) Consider a continuous time MC with state space S = {1, 2, 3} and infinitesimal matrix

Find the stationary distribution

(e) In (d), find P (reach 2 before 1|X0 = 3)

6.Consider a branching process with offspring distribution Z ∼ Bin(3, 0.7). Let Xn denote the population at time n, with X0 = 1. Find,

(a) P (X1 = 2|X2 = 5)

(b) P (X2 = 0|X3 = 0)

(c) Var (X4|X2 = 3)

(d) limn→∞ P (Xn = 0) (e) (5 points) P (X4 ≥ 1|X3 ≥ 1)

7. 

(a) Consider 3 independent Poisson process with respective intensities 1, 2, 3. Define W to be the first time that all 3 processes each have at least one event occurring. Thus W = min{t : Ni(t) ≥ 1, i = 1, 2, 3}. Find E(W)

(b) Consider a Markov chain with

Find the probability that starting in state 1, that the chain eventually gets absorbed into state 3.

(c) Claims arrive to an insurance company according to a Poisson process with rate λ = 5 claims per day. The claim sizes are iid with pdf

f(x) = (0.003)2xe−0.003x , x > 0

Find the probability that there will be at least 4 claims of size $1000 or greater over a 3 day period

(d) Suppose that X, Y, Z are independent exponential random variables with E(X) = 1, E(Y ) = 2 and E(Z) = 3 (Thus X ∼ Expo(1), Y ∼ Expo(1/2), Z ∼ Expo(1/3). Find

(e) Consider a M/G/∞ queue in which service times have pdf,

The arrivals are from a Poisson process with λ = 5. Find the distribution of the number of customers in service at time 3, who arrived to the system during 2.1 < t < 2.7