Instructions: You may only consult the textbook and lecture notes: do not consult any

other materials or other persons. Reference any results you use from the textbook or lecture

notes. Solutions must be produced using LaTeX and submitted as hardcopy. Handwritten

solutions will not be accepted.

1) Let S and T be independent random variables that each have the uniform distribution on

the interval [−1, 1]. Put R =

√

S2 + T

2

.

Show that conditional on the event {R ≤ 1} the two random variables

X :=

S

R

p

−2 log R2 and Y :=

T

R

p

−2 log R2

are independent standard normal random variables.

[10 points]

2) Let X and Y be independent nonnegative random variables with density functions f and

g that are continuous on (0, ∞). Suppose for any u > 0 that the conditional distribution of

X given X + Y = u is uniform on the interval [0, u].

Show that X and Y are identically distributed and that their common distribution is exponential.

[10 points]

Hint: Write h for the density function of X + Y . Explain why the density function of

(X, X +Y ) is given by fX,X+Y (x, u) = 1

u

h(u) for 0 ≤ x ≤ u. Then explain why 1

x+y

h(x+y) =

f(x)g(y) for x, y ≥ 0. By considering what happens when x = 0 and y = 0, show that f = g.

Conclude that f(x+y)f(0) = f(x)f(y) for x, y ≥ 0. Define a function ¯f by ¯f(z) = f(z)/f(0).

Show that ¯f(x + y) = ¯f(x)

¯f(y) for x, y ≥ 0 . Conclude that ¯f(z) = e

−λz when z ≥ 0 for

some λ > 0. Lastly, observe that f(x) = λe−λx when x ≥ 0.

3) Given two n×n matrices A = (aij ) and B = (bij ), their Schur product is the n×n matrix

C = (cij ) with i, j entry cij = aij bij for 1 ≤ i, j ≤ n. Suppose that Σ0 and Σ00 are two n × n

nonnegative definite matrices with Schur product Σ.

Show that Σ is also nonnegative definite.

[10 points]

1

Hint: Explain why we may assume that there are independent random vectors X0 and X00

such that X0 and X00 both have mean vector 0, X0 has variance-covariance matrix Σ

0

, and

X00 has variance-covariance matrix Σ

00. Then consider the variance-covariance matrix of the

random vector X, where Xi = X0

iX00

i

for 1 ≤ i ≤ n.

4) Alice and Bob are having a snowball fight. The times at which Alice throws her snowballs

are the arrival times of Poisson process with (constant) intensity λ. The times at which Bob

throws his snowballs are the arrival times of Poisson process with (constant) intensity µ.

Assume that these two Poisson processes are independent. Consider the event that in the

time interval [0, T] Alice and Bob throw the same non-zero number of snowballs and that

they alternate throws (for example, one way this could happen is that they each throw two

snowballs and the sequence of throws is Alice, Bob, Alice, Bob).

What is the conditional probability given this event that Alice throws the first

snowball?

[10 points]

Hint: Use the coloring theorem to build the Poisson processes for Alice and Bob from a single

Poisson process with intensity λ + µ.

5) Suppose that the random variable U has the uniform distribution on [0, 1] and that

conditional on U = u the random variable X has the binomial distribution with number of

trials n and success probability u.

Show using generating functions or otherwise that P{X = x} =

1

n+1 for x ∈

{0, 1, . . . , n}.

[10 points]

6) Let 0 = S0, S1, S2, . . . be a simple random walk on the integers with p = q =

1

2

. Recall

that p0(n), n ≥ 0, is the probability that the walk is at position 0 at time n and f0(n),

n ≥ 1, is the probability that the walk first returns to position 0 at time n. We have the

generating functions P0(s) = P∞

n=0 p0(n)s

n =

P∞

k=0 p0(2k)s

2k and F0(s) = P∞

n=1 f0(n)s

P

n =

∞

k=1 f0(2k)s

2k

. Put a(0) = 1, a(2k) = P{S1 6= 0, S2 6= 0, . . . , S2k 6= 0} for k ≥ 1.

Show that a(2k) = p0(2k) =

2k

k

(

1

2

)

2k

for k ≥ 0.

[10 points]

Hint: Explain why a(2k) = P∞

`=k+1 f0(2k). Put A(s) = P∞

k=0 a(2k)s

2k and show that

A(s) = 1 − F0(s)

1 − s

2

.

7) Let 0 = S0, S1, S2, . . . be a simple random walk on the integers with 0 < p < q < 1.

Say that the time n ≥ 0 is an upcrossing time for the random walk if Sn < Sn+1 (that is

Sn+1 = Sn + 1). Let T−1 = min{n ≥ 0 : Sn = −1} be the first time that the random walk

visits −1.

Find the probability generating function for the number of upcrossing times that

occur before time T−1.

[10 points]

Hint: Put Z0 = 1 and define Zm for m ≥ 1 by

Zm = |{0 ≤ n < T−1 : Sn = m − 1, Sn+1 = m}|;

that is, Zm for m ≥ 1 is the number of times the random walk makes an upcrossing from

state m − 1 to state m prior to the time T−1. Show that Z0, Z1, Z2, . . . is a branching process

with P{# of offspring = k} = p

k

q for k ≥ 0.

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