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Student Number
Queen’s University
Department of Mathematics and Statistics
STAT/MTHE 353
Final Examination April 21, 2012
Instructor: T. Linder
• PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation
of exam questions. Do your best to answer exam questions as written.”
• This material is copyrighted and is for the sole use of students registered in MTHE/STAT
353 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may also
constitute a breach of academic integrity under the University Senate’s Academic
Integrity Policy Statement.
• Formulas and tables are attached.
• An 8.5 × 11 inch sheet of notes (both sides) is permitted.
• Simple calculators (Casio 991, blue or gold sticker) are permitted. HOWEVER, do
reasonable simplifications.
• Write the answers in the space provided, continue on the backs of pages if needed.
• SHOW YOUR WORK CLEARLY. Correct answers without clear work showing
how you got there will not receive full marks.
Marks: Please do not write in the space below.
Problem 1 [10]
Problem 4 [10]
Problem 2 [10]
Problem 5 [10]
Problem 3 [10]
Problem 6 [10]
Total: [60]
STAT/MTHE 353 – Final Exam, April 21, 2012
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1. Let X1 , X2 , and X3 be independent exponentially distributed random variables with
parameter 1. Define Y1 = X1 + X2 , Y2 = X1 + X3 , and Y3 = X2 + X3 .
(a) Find the joint probability density function of Y1 , Y2 , Y3 .
[6]
STAT/MTHE 353 – Final Exam, April 21, 2012
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(b) Find the probability P (Y1 > Y2 > Y3 ). Hint: Do not integrate the joint pdf of
Y1 , Y2 , Y3 .
[4]
STAT/MTHE 353 – Final Exam, April 21, 2012
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2. Suppose we are given a random N number of balls, where N is a Poisson random variable
with parameter λ. We distribute these N balls randomly into n urns in such a way that
each ball is equally likely to be placed in any of the n urns, independently of the placement
of the other balls. Compute the expected number of empty urns.
[10]
STAT/MTHE 353 – Final Exam, April 21, 2012
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3. Suppose that at a dinner party attended by n people, each of the n2 pairs of people
are, independently, friends with probability p. Let Xi , i = 1, . . . , n denote the number of
friends at the dinner party of the ith individual.
(a) What is the distribution of Xi ?
[2]
(b) Find ρ(Xi , Xj ), the correlation coefficient between Xi and Xj for i 6= j.
[8]
STAT/MTHE 353 – Final Exam, April 21, 2012
4. (a) Let X and Y be random variables having finite variances. Show that
Cov X, E[Y |X] = Cov(X, Y ).
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[5]
(b) Suppose that Var(X) > 0 and for some real constants a and b, E[Y |X] = a + bX.
Cov(X, Y )
.
[5]
Show that b =
Var(X)
STAT/MTHE 353 – Final Exam, April 21, 2012
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5. (a) Suppose that the mean lifetime of a certain type of cheap light bulb is 100 hours
with a standard deviation of 30 hours. You light a room with a lamp that uses this type
of light bulb, and when a light bulb fails, you immediately replace it. Use the central
limit theorem to estimate how many of these light bulbs you should store so that the
probability that you can continually light the room for the next 2000 hours is at least
0.95.
[6]
STAT/MTHE 353 – Final Exam, April 21, 2012
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(b) Let Y1 , Y2 , . . . be a sequence of independent Bernoulli(1/2) random variables and
define Xn = 2n Y1 Y2 · · · Yn . Does Xn converge in probability, and if yes, to what limit?
[4]
Bonus: Does Xn converge almost surely, and if yes, to what limit?
STAT/MTHE 353 – Final Exam, April 21, 2012
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6. Let Y1 , Y2 , . . . be independent and identically distributed discrete random variables taking
values in {0, 1, . . . , 9} with probabilities P (Yi = k) = 1/10 for k = 0, 1, . . . , 9. Let X be a
continuous random variable that is uniformly distributed on the interval [0, 1].
(a) Find the moment generating function MYi (t) of the Yi ’s and the moment generating
function MX (t) of X.
[4]
STAT/MTHE 353 – Final Exam, April 21, 2012
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P
(b) Let Xn = ni=1 10−i Yi . Use the Lévy continuity theorem for moment generating
functions to show that Xn converges to X in distribution as n → ∞.
[6]
STAT/MTHE 353 – Final Exam, April 21, 2012
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Formula Sheet
Sums
∞
X
xk =
k=0
1
, if |x| < 1,
1−x
N
X
xk =
k=0
1 − xN +1
if x 6= 1.
1−x
Special Distributions
P (X = 0) = 1 − p, P (X = 1) = p
E(X) = p, Var(X) = p(1 − p).
Bernoulli(p):
Binomial(n, p) :
Geometric(p):
n k
P (X = k) =
p (1 − p)n−k , k = 0, 1, 2, . . . , n,
k
E(X) = np, Var(X) = np(1 − p).
P (X = k) = p(1 − p)k−1 , k = 1, 2, . . . .
E(X) = 1/p, Var(X) = (1 − p)/p2 .
λk −λ
e , k = 0, 1, 2, . . . ,
k!
E(X) = λ, Var(X) = λ.
Poisson(λ):
P (X = k) =
Continuous uniform on [a, b]:

1/(b − a) if a ≤ x ≤ b
f (x) =
0
otherwise,
E(X) =
a+b
,
2
Var(X) =
(b − a)2
.
12
Exponential:
f (x) =

λe−λx
if x ≥ 0
0
otherwise.
E(X) = 1/λ,
Var(X) = 1/λ2 .
Normal (Gaussian) with mean µ and variance σ 2 :
Z x
(x−µ)2
(t−µ)2
1
1
−
2
f (x) = √ e 2σ
and F (x) = √
e− 2σ2 dt.
σ 2π
σ 2π −∞
Standard Normal (Gaussian) with µ = 0 and σ 2 = 1:
Z z
t2
1 − z2
1
f (z) = √ e 2 , and F (z) = Φ(z) = √
e− 2 dt.
2π
2π −∞
STAT/MTHE 353 – Final Exam, April 21, 2012
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The distribution function, Φ(z), of a standard normal random variable
Note: Φ(−z) = 1 − Φ(z) for any z ∈ R.