FRE6083, Midterm Examination, Wednesday March 12 2014, 5:30pm-8:00pm
1. Number of pages including this one: 2 2. For this examination, you may only use a 2-page cheat sheet. No other notes, books or electronic devices may be used. Cell phones may not be used. Any violation of this policy will result in a grade of zero for this exam. Problem 1 (12 points) We consider X1 , X2 , · · · Xn , independent and identically distributed random variables with common mean µ ∈ R and common variance σ 2 > 0. 1. (4 points) What is the limit of
1 n n i=1

Xi as n → +∞? Justify your answer.

We suppose that µ is not known. You observe 1000 samples drawn from the common distribution of Xi and you estimate the mean µ by using the sample mean estimator 1 x= 1000
1000

Xi . i=1 2. (4 points) Give the approximate distribution of the error x − µ for the above estimate. 3. (4 points) In particular, give the approximate mean and variance of x − µ. Problem 2 (16 points) We consider a discrete time and discrete space Markov chain Xn with state space {0, 1, 2, 3} that has the one-step transition probability matrix   1/3 0 2/3 0 1/2 0 0 1/2  P =  0 1/3 1/3 1/3 0 1/3 0 2/3 and assume that X0 = 0. 1. (5 points) Is this Markov chain irreducible? Justify your answer. 2. (3 points) Compute P (2) . 3. (4 points) Compute E[X2 |X0 = 0]. 4. (4 points) What is the probability that the chain will move to the state 1 for the first time after exactly 2 steps?

1

Problem 3 (22 points) Consider the stochastic process X(t) = e−Y t , for t > 0, where Y is a random variable with a uniform distribution on the interval (0, 1). 1. (6 points) Calculate the first-order density function of the process {X(t), t > 0} 2. (5 points) Compute E[X(t)], for t > 0. 3. (5 points) Compute the auto-covariance function CX (t, t + s) for s, t > 0. 4. (6 points) Is the process X wide sense stationary? Justify your answer. Is it strict sense stationary? Justify your answer. Problem 4 (23 points) Consider the loss function
N (t)

L(t) = i=1 e−rτi Xi ,

where N (t) is a Poisson process with parameter λ > 0, X1 , X2 , · · · are independent and identically distributed random variables with common mean µ ∈ R and variance σ 2 > 0 and τi is the time of the ith claim. Here N (t) models the cumulative number of insurance claims that have been made during the time period [0, t] and Xi represents the loss incurred by the ith claim. Furthermore, to simplify, we assume that the claim amounts X1 , X2 , · · · are independent of the claim arrival times τ1 , τ2 , · · · . Finally, you will need to use the following well-known result: conditional on N (t) = n, the variables τ1 , τ2 , · · · , τn are distributed as the ordered values of n independent uniform random variables in (0, t). 1. (4 points) Compute E[ n −rτi |N (t) i=1 e

= n].

2. (4 points) Compute E[L(t)|N (t) = n]. 3. (5 points) Use the previous result and the total probability rule to compute E[L(t)]. 4. (20 points) Compute var[L(t)]. Problem 5 (17 points) We model the growth of a population of bacteria by using a pure birth process denoted by X(t). When there are n individuals in the population, the average time needed for a birth to occur is 1/n hours for n ≥ 0 1. (4 points) Compute p1,1 (t). 2. (6 points) Compute p1,2 (t) by using the recursive formula seen in class for j > i: t pi,j (t) = λj−1 exp(−λj t)
0

exp(λj s)pi,j−1 (s)ds.

3. (7 points) Knowing that at time 0, there is one individual in the population, what is the probability that there will be exactly 1 birth during the first 2 hours? 2