Applied ProbabilitySpring 2015

Take Home Final ExamDue: Friday May 1, High Noon (in my mailbox)

Directions: Show all work to receive possible partial credit. Unsupported guesses willmeet a red pen.

This test is to be YOUR OWN WORK. You are allowed the use your textbook and coursenotes only. You are not allowed to talk to other students about it, work together, or askprevious applied probability students answers. I ask for your compliance on this matter!

1. Meteors strike the moon according to a Poisson process with arrival rate � > 0. Thesize of the individual meteors are independent and identically distributed with cumulativedistribution function G. Starting from time zero, how long does one wait, on the average,for a meteor of size x or larger to strike the moon?

2. Suppose that X1, . . . , X5 is an IID sample from a continuous distribution. What is thechance that X1 is the smallest of the five draws? What is the chance that the five drawsare made in increasing order?

3. What is E[max(U1, . . . , Un)] where U1, . . . , Un are IID Unif[0, t] random variables? Nowsuppose you have a Poisson process with arrival rate � > 0. At time t, how long has itbeen, on the average, since the last event occurred?

4. For a Markov chain {Xn}1n=0, show that

P

(n)i,j =

nX

k=0

f

(k)i,j P

(n�k)j,j .

5. Consider consecutive independent rolls of a fair six sided die. Let Xn be the numberof times the last roll has appeared consecutively. For example, if n = 5 and the five rollswere 2, 4, 5, 4, 4, then (X1, X2, X3, X4, X5) = (1, 1, 1, 1, 2). Is {Xn} an irreducible aperiodicMarkov chain? If so, what is its limiting distribution?

6. For a Markov chain {Xn}, show that

P (Xn = in|Xn�2 = in�2 \Xn�1 = in�1 \Xn+1 = in+1 \Xn+2 = in+2)

and

P (Xn = in|Xn�1 = in�1 \Xn+1 = in+1).

are one and the same.

In Chapter 5, work problems 12, 34, 42, 46, 50, 52,78