Math 180C (P. Fitzsimmons) Second Midterm Exam November 18, 2022
Name PID
This exam is closed book and closed notes.
You may use a calculator and your formula sheet.
Show your work and justify your answers.
1. [25 points] Airplanes arrive at Berghlind Field at the times of a renewal process {N(t) : t ≥ 0} with
mean interarrival time equal to µ. The number of passengers on plane k is a random variable Ck, with
mean ν = E[Ck]. Assume that C1, C2, . . . are independent and identically distributed, and independent of
the plane-arrival process. As usual, let M(t) = E[N(t)], and let
Z(t) =
N(t)∑
k=1
Ck, t ≥ 0,
denote the total number of passengers that have arrived by time t.
(a) Find an expression for E[Z(t)] in terms of M(t) and ν.
1
(b) Use the renewal theorem to find
lim
t→∞
E[Z(t)]
t
,
the long-term arrival rate of passengers.
2
Name PID
2. [25 points] Let {N(t) : t ≥ 0} be a renewal process with arrival times W1,W2, . . .. Suppose that the
renewal function M is given by the formula
M(t) =
3t
2
− 1
4
· (1− e−6t) , t ≥ 0.
(a) Find µ, the mean inter-arrival time.
(b) Find E[WN(t)+1].
3
(c) Find limt→∞ E[γt], where γt is the excess life process associated with the renewal process.
4
Name PID
3. [25 points] Consider a renewal process for which the inter-arrival times X1, X2, . . ., have density function
f . Assume that each Xk has finite mean µ and finite variance σ
2. Recall that βt = WN(t)+1 −WN(t) is the
total length of the renewal cycle in progress at time t. (Convention: W0 = 0.) Let H(t) = E[βt] for t > 0.
(a) Explain why H satisfies the renewal equation
H(t) = h(t) +
∫ t
0
H(t− s)f(s) ds, t > 0,
where h(t) = E[X1 · 1{X1>t}] =
∫∞
t
xf(x) dx.
5
(b) Verify that
∫∞
0
h(t) dt = σ2 + µ2.
(c) Use the results of parts (a) and (b), and the Renewal Theorem, to find
lim
t→∞E[βt].
6
Name PID
4. [25 points]. Let {B(t) : t ≥ 0}, be a standard Brownian motion. Fix u > 0 and define a stochastic
process {X(t) : t ≥ 0} by the formula
X(t) = B(t+ u)−B(u), t ≥ 0.
(a) Find E[X(t)].
b) Find Cov[X(s), X(t)].
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(c) Explain why {X(t) : t ≥ 0} is a standard Brownian motion.