STAT 416 SAMPLE FINAL EXAMINATION Fall, 2020
Problem 1 (midterm 1)
The joint probability density function (p.d.f.) of the random variables (X, Y ) is
f(x, y) =
(x2 − y2)e−x
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, 0 < x <∞,−x ≤ y ≤ x
1. Derive fX(x), the marginal p.d.f. of X (please be sure to specify the range of x).
2. Derive fY |X=x(y), the conditional p.d.f. of Y , given that X = x.
3. Compute E(Y 3|X = x).
Problem 2 (midterm 1)
A hospital admits two types of patients on any day: emergency care patients and non-emergency
care patients. Let X denote the total number of emergency care patients admitted to the hospital on
any day where X ∼ Poisson(λe). Let Y denote the total number of non-emergency care patients
admitted to the hospital on any day where Y ∼ Poisson(λne). Assume X and Y are independent,
1. Derive the distribution of the total number of patients admitted to the hospital on any given
day? (You need provide all the steps, name the distribution and its relevant parameters).
2. What is the probability of admitting k emergency care patients on a given day given that a
total number of n patients (where n ≥ k) were admitted to the hospital on that day? (Need
to provide all the steps on how you have calculated this probability). Specify this particular
distribution.
Problem 3 (midterm 1)
A coin comes up heads with probability p is continually flipped until the pattern Tails, Heads,
Tails appear (That is, you stop flipping when in the most recent three flips, the first and last flips
land Tails and the second flip lands Heads). Let X denote the number of flips made. Find E(X).
Problem 4 (midterm 2)
Let X1, X2, and X3 be iid exponential random variables with rate λ = 1. Let Y1 and Y2 be iid
exponential random variables with rate λ = 0.5, and Z1 and Z2 be iid exponential random variables
with rate λ = 2. All X , Y , and Z random variables are independent of each other.
1. What is the expected value and variance of min{X1, X2, X3, Y1, Y2, Z1, Z2}? (make sure
you specify the distribution)
2. What is the probability that max{Y1, Y2, Z1, Z2} > 1?
Problem 5 (sample midterm 2)
A taxicab driver moves between the airport A and two hotels, B and C, according to the follow-
ing rules. If he is at the airport, then he will go to one of the two hotels next with equal probability.
If he is at one of the hotels, then he will go to the airport with probability 3/4 and to the other hotel
with probability 1/4.
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1. Write down the Transition Probability Matrix.
2. Suppose the driver starts at the Hotel B at time 0. Find the probability for each of his possible
three locations at time 2.
3. Show that the limiting distribution for this Markov Chain exists (i.e. Go through the check-
list).
4. Compute the limiting distribution using the long run proportions (or ergodic) equations.
5. Compute the limiting distribution using the detailed balance equations.
Problem 6
Suppose that a popcorn machine produces popcorn kernels according to a Poisson process with
a rate of 2 per minute.
1. What is the probability that exactly two popcorn kernels are produced in the interval from 0
to 4 minutes?
2. Conditioned on there being exactly three popcorn kernels produced in the first 4 minutes,
what is the probability that there is exactly one in the first minute?
3. Find the expected number of minutes until 5 popcorn kernels are produced.
Problem 7
Let N(t) be a counting process which is a generalization of a Poisson process, where:
N(t)−N(s) ∼ Pois(λ(t− s))
and λ itself is an exponential random variable with rate 2:
λ ∼ Exp(2)
1. Calculate the expected number of events in the time interval (0, 3).
2. Calculate the variance of the number of events in the time interval (0, 3).
Problem 8
Jill picks apples at times of a Poisson process with rate 2 per hour.
1. If 40% of the apples are ripe and 60% are unripe, what is the probability she will pick exactly
1 ripe and 2 unripe applies if she picks for 2.5 hours?
2. Jill will pick apples until she is tired. Suppose that she gets tired according to a uniform
distribution from 0 to 2 hours. Find the variance of the number of apples she picks by the
time she gets tired.
Problem 9
People arrive to a bus stop according to a Poisson process with rate λ. Buses arrive according
to a Poisson Process (independent of the arrival of people) at a rate of µ.
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1. Find the mean number of people waiting at the time of the arrival of the first bus.
2. Find the variance of the number of people waiting at the time of the arrival of the first bus.
Problem 10
Suppose birds arrive at the PSU Arboretum according to a Poisson process with rate λ (e.g.
{N(t), t ≥ 0}). Suppose counts are taken at specific times T , which is a uniform random variable
with range (0, 100) (Hint: What is the mean and variance of this random variable?). Let N(t) be
independent of T . Find:
1. Covariance between the times T and the count of birds N(T )?
2. Variance of N(T )?
Problem 11
There are two bowling lanes in the bowling alley. Pairs of players arrive at rate 2 per hour
and bowl for an amount of time that is exponentially distributed with mean 1 hour. If two pairs of
players are already playing, the new players will leave the bowling alley. Express the number of
pairs bowling at the bowling alley as a Markov chain using the rates at which the process makes a
transition when in a particular state (vi’s) and the probabilities of transitioning when you are leav-
ing a particular state, you next enter a different state (the Pij).
Problem 12
Suppose a restaurant has two wood-fired pizza ovens and two pizzaiolos (pizza making profes-
sionals). Each oven can only bake one pizza at a time. Each pizzaiolo can only prepare one pizza
at a time, and the average preparation time is 2 minutes. The pizzaiolos will begin prepping a pizza
when there is an empty oven. The average time for the pizza to fully cook in the wood-fired oven
is 10 minutes.
1. In the long-term, what is the proportion of time that there are zero pizzas in the oven?
2. In the long-term, what is the proportion of time when two pizzas are in the oven?
Problem 13
An airline phone-reservation system is called a call center and is staffed by s reservation clerks
called agents. An incoming call for reservations is handled by an agent if one is available; other-
wise the caller is put on hold. The system can put a maximum of H callers on hold.When an agent
becomes available, the callers on hold are served in order of arrival. When all the agents are busy
and there are H calls on hold, any additional callers get a busy signal and are permanently lost. Let
X(t) be the number of calls in the system, those handled by the agents plus any on hold, at time t .
Assume the calls arrive according to a Poisson Process with rate λ and the processing times of the
calls are iid Exp(µ) random variables. Model {X(t), t ≥ 0} as a CTMC. Provide the generator
matrix for s = 2 and H = 3.
Problem 14
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Consider two machines. Machine i operates for an exponential time with rate λi and then fails;
its repair time is exponential with rate µi, i = 1, 2. The machines act independently of each other.
Define a four-state continuous-time Markov chain that jointly describes the condition of the two
machines (clearly describe the states, the rates (vi’s), and the jump probabilities (Pij)).
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