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Student Number
Queen’s University
Department of Mathematics and Statistics
STAT 353
Final Examination April 24, 2010
Instructor: G. Takahara
• “Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam questions as written.”
• “The candidate is urged to submit with the answer paper a clear statement of
any assumptions made if doubt exists as to the interpretation of any question that
requires a written answer.”
• Formulas and tables are attached.
• An 8.5 × 11 inch sheet of notes (both sides) is permitted.
• Simple calculators 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 per part question are shown in brackets at the right margin.
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 353 -- Final Exam, April 24, 2010
1.
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Let X and Y be independent U (0, 1) random variables. Suppose we form a circle of
radius X and a rectangle with width X and height Y .
(a) Find the probability that the area of the rectangle is greater than the area of the
circle.
[5]
STAT 353 -- Final Exam, April 24, 2010
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(b) Find the correlation coefficient between the area of the rectangle and the area of the
circle.
[5]
STAT 353 -- Final Exam, April 24, 2010
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2. Each card in a deck of 52 playing cards is one half of a pair, where the other card in the
pair has the same colour and value. Thus there are 26 pairs in a deck. Suppose the cards
are drawn one at a time and each card, independently from card to card, is either placed
into a bag with probability 1/3 or is destroyed with probability 2/3. Find the expected
number of intact pairs in the bag.
[10]
STAT 353 -- Final Exam, April 24, 2010
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3. Let N , X1 , X2 , . . . be independent and identically distributed Geometric(p) random variables. Find the moment generating function of N and of Y = X1 + X2 + . . . + XN .
[10]
STAT 353 -- Final Exam, April 24, 2010
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√
4.(a) Let Xn have a Gamma(n,1) distribution, for n ≥ 1. Find limn→∞ P (Xn > n + 2 n).
[6]
√
√
(b) Let Yn have a Gamma(n, n) distribution, for n ≥ 1. Find limn→∞ P (Yn > n+2 n).
[4]
STAT 353 -- Final Exam, April 24, 2010
5.
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Let Y1 , Y2 , . . . be independent and identically distributed random variables uniformly
distributed on the interval [a, b], where a < b. For n ≥ 1, let Xn = min(Y1 , . . . , Yn ).
(a) Show that Xn converges to a almost surely as n → ∞.
[4]
STAT 353 -- Final Exam, April 24, 2010
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(b) If a > 0, show that P (np Xn > M ) → 1 as n → ∞ for any fixed p > 0 and M > 0,
while if a = 0 show that nXn converges in distribution to X as n → ∞, where X has an
exponential distribution with parameter 1/b.
[6]
STAT 353 -- Final Exam, April 24, 2010
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6. At time t = 0 the price of a stock is 50 dollars/share. At time t = 1 it will be either
25 or 150 dollars/share. At time 0 we can “buy” x shares of stock and y shares of an
option (negative x or y values means we sell). Each share of the option we hold allows
us to buy 1 share of the stock at 125 dollars/share. The price of the option at t = 0 is c
dollars/share. Assume there is no discount factor (i.e., α = 0).
(a) Write the returns for each wager (the two wagers are x shares of stock and y shares
of option) for each possible outcome at time t = 1 and find a value of c and a distribution
on the outcomes at time t = 1 so that no sure win is possible.
[5]
STAT 353 -- Final Exam, April 24, 2010
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(b) Give a betting strategy (x, y) that produces a sure win if c is greater than the value
determined in part(a). Hint: Give a betting strategy that makes the return the same no
matter what the price of the stock is at time t = 1.
[5]
STAT 353 -- Final Exam, April 24, 2010
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Formula Sheet
Special Distributions
Continuous uniform on (a, b):

1/(b − a) if a ≤ x ≤ b
f (x) =
0
otherwise,
Gamma with parameters r and λ:

 λr xα−1 e−λx
f (x) = Γ(r)
0
E[X] =
if x > 0
otherwise.
a+b
,
2
E[X] =
Var[X] =
r
,
λ
(b − a)2
.
12
Var[X] =
r
.
λ2
Normal (Gaussian) with mean µ and variance σ 2 :
Z x
(x−µ)2
(t−µ)2
1
1
−
and F (x) = √
e− 2σ2 dt.
f (x) = √ e 2σ2
σ 2π
σ 2π −∞
Geometric with parameter p:

p(1 − p)k−1
P (X = k) =
0
k = 1, 2, . . . ,
otherwise.
Binomial with parameters n and p:
  n pk (1 − p)n−k k = 0, 1, . . . , n
k
P (X = k) =
0
otherwise.
1
E[X] = ,
p
E[X] = np,
Var[X] =
1−p
.
p2
Var[X] = np(1 − p).
• df and pdf of the kth order statistic from a random sample X1 , . . . , Xn :
Fk (x) =
n X
n
i=k
fk (x) =
i
[F (x)]i [1 − F (x)]n−i ;
n!
f (x)[F (x)]k−1 [1 − F (x)]n−k ,
(k − 1)!(n − k)!
where F (x) and f (x) are the df and pdf, respectively, of each Xi .
STAT 353 -- Final Exam, April 24, 2010
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The distribution function of a standard normal random variable