University of Western Ontario
Departments of Applied Mathematics
Statistics 2503B Final Examination
(Take-Home)
Code 111
April 22, 2020 24 hours
Student’s Name: Student Number:
Instructions
1. Print your Name, Student Number in the box above.
2. The Exam Booklet should have 20 pages (including the front page).
3. In Part A (Multiple Choice questions), circle the correct answer
for each multiple choice question.
4. Part B must be answered in the space provided in the Exam Booklet.
Unjustified answers will receive little or no credit.
5. Pages 18, 19 and 20 of the Exam Booklet are blank and are to be
used for Part B if you need extra space for presenting your answers
for Part B. Indicate clearly which questions from Part B you are
answering there.
6. Code of Conduct: Students are not allowed to assist or commu-
nicate to each other during the exam time. This will constitute a
scholastic offence subject to severe academic penalties. If there is any
indication of violation detected, it will result a ZERO to the exam.
7. Total Marks = Part A (30) + Part B (90) = 120 marks.
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Part A: 15 multiple choice questions (2 marks each) = 30 marks
Do your working in the Scratch Papers. Circle the correct
answer for each multiple choice question.
A1: Adam has $1,000 in the bank right now, in an account which pays 5 %
interest (compounded continuously). At what (continuous) rate, rounded
to the nearest dollar, does Adam need to deposit money into the account
in order to have $20,000 after 20 years?
A) $582 per year B) $503 per year
C) $318 per year D) $101 per year
E) none of the above
A2: The Wronskian of y1 = e−2t cos t and y2 = e−2t sin t is
A) e2t(sin t+ cos t) B) e4t
C) e−4t D) e−2t(cos t+ sin t)
E) none of the above
A3: The general solution to the differential equation y′′ − 4y′ + 5y = 0 is
A) c1e2x cosx+ c2e2x sinx B) c1 sin 2x+ c2 cos 2x
C) c1ex cos 2x+ c2ex sin 2x D) c1 cosx+ c2 sinx
E) none of the above
A4: The inverse Laplace transform of F (s) = s
s2 − 4s+ 5 is
A) 2e2t cos t+ e2t sin t B) e2t(3 cos t− 5 sin t)
C) e2t(cos t+ 2 sin t) D) e2t cos 2t+ 5e2t
E) none of the above
A5: The inverse Laplace transform of F (s) = ln
(
s− 2
s+ 2
)
is
A) t
2
(t2 + 1)4
B) e
−2t − e2t
t
C) t− 4e2t D) e
2t + e−2t
t
E) none of the above
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A6: The solution to the difference equation 3y(n+ 1) = 2y(n) + 1, y(0) = 3 is
A) 2
n+1
3n
+ 1 B) 3n + 2
C) 2(2n−1 + 1) D) 3(3
n + 1)
2n+1
E) none of the above
A7: A Markov chain has a state space S = {1, 2} and transition probability ma-
trix P . In the long run, the chain spends twice as much time in state 1 as it
does in state 2. When n is large, we would expect P n to be approximately
equal to
A)
 1/3 2/3
1/3 2/3
 B)
 2/3 2/3
1/3 2/3

C)
 2/3 2/3
1/3 1/3
 D)
 2/3 1/3
2/3 1/3

E) none of the above
A8: The differential equation dy
dt
= (y − 1)(3 − y) has y = 1 and y = 3 as its
equilibrium solutions. Then
A) y = 3 is a stable solution B) lim
t→∞
y(t) = 1 if y(0) = 1/2
C) y = 1 is an unstable solution D) both (A) and (C)
E) none of the above
A9: Given y1 = x is a solution of the Riccati equation y′ = 1 − 2xy + x2 + y2
then the general solution is
A) x+ 1
c− x B) x+
x
c+ x
C) x+ e
x
c− x D) x+
ex
c− ex
E) none of the above
A10: L−1
{
1
(s2 + 1)2
}
=
A) 1
2
t sin t− 1
2
cos t B) 1
2
sin t− 1
2
t cos t
C) 1
2
t sin t+
1
2
cos t D) 1
2
sin t+
1
2
t cos t
E) none of the above
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A11: Choose the differential equation which has the following direction field.
-4 -2 0 2 4
-4
-2
0
2
4
x
y
A) y′ = x+ xy B) y′ = 2− y
C) y′ = x+ y − 1 D) y′ = y2 − x2
E) y′ = x(2− y)
A12: A particular solution of y′′ + y′ + y = 2x+ 1 is
A) y = x− 2 B) y = 2x+ 2
C) y = 2x− 1 D) y = 2x+ 1
E) none of the above
A13: The general solution to the difference equation y(n+2)+y(n+1)−6y(n) = 0
is
(A) c1(−2)n + c2 · 3n (B) c1 · 2
n+1
3n
+ c2
(C) c1 · 2n + c2 · 3n (D) c1 · 2n + c2(−3)n
A14: The following is the transition probability matrix of a Markov chain with
state space {0, 1, 2, 3}.
P =

0 1/2 0 1/2
0 0 1 0
1 0 0 0
0 0 1 0

The period of state 0 is
(A) aperiodic (B) 2 (C) 3 (D) 4 (E) none of the above
A15: The solution of the differential equation x2y′ − y = 2x3e−1/x is
(A) y = Ce−2x (B) y = e−1/x(x2 + C) (C) y = (xex − ex + C)/x
(D) y = e1/x(x2 + C) (E) y = Ce2x + x2
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Part B: Show all your work for each of the following questions.
Total: 90 marks. Do all the 10 questions between B1 and B10.
B1: (24 marks) Solve the following initial value problems. When it is possible,
express the solutions explicitly in terms of the independent variable.
a) (t2 + 1)dy
dt
+ 3ty = 6t, y(0) = −1
b)dy
dt
= y(1− y), y(0) = 1/2
c) dy
dt
+
1
t
y = ty2, y(1) = 1
d) d
2y
dt2
− 4y = 3e2t, y(0) = −2, y′(0) = 0
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B1 (continued)
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B2: (6 marks) Consider the differential equation dy
dx
=
x2 + y2 − 5
y + xy
.
(a) Check if the differential equation is exact. If not, find an integrating
factor.
(b) Using the integrating factor in (a) to solve the above equation subject
to the initial condition y(0) = 1.
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B3: (6 marks) Use the method of variation of parameters to obtain a particular
solution of
y′′ + 4y = sec 2t
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B4: (6 marks) Find L−1
{
1
(s2 + 4)2
}
.
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B5: (8 marks) Use the method of eigenvalue to solve the following initial-value
problem
x′ =
 0 1
−4 −4
x, x(0) =
 x(0)
y(0)
 =
 1
0

where x =
 x(t)
y(t)
.
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B6: (8 marks) Use the method of Laplace transform to solve the initial value
problem.
dx
dt
= y(t) + 2
dy
dt
= −4x(t)− 4y(t) + t
subject to the initial condition x(0) = 1, y(0) = 0.
Hint: Let X(s) = L{x(t)}, Y (s) = L{y(t)}. Form a system of two equa-
tions in two unknown X(s) and Y (s) then use Cramer’s to obtain X(s) and
Y (s). After that use inverse Laplace transform to obtain x(t) and y(t).
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B6 (continued)
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B7: (6 marks) Let X be a discrete random variable with the mass density func-
tion
p(x) =
 x− 1
k − 1
 pk(1− p)x−k, x = k, k + 1, k + 2, . . .
(a) Find MX(t).
(b) Using MX(t), find E(X), E(X2) and V ar(X).
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B7 (continued)
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B8: (6 marks) Let X be a continuous random variable with the probability
density function
f(x) =
 32x
(
1− x
2
)
, 0 ≤ x ≤ 2
0, otherwise
(a) Find MX(t).
(b) Using MX(t), find E[X], E[X2] and V ar(X).
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B9: (8 marks) Consider the Markov chain with states 0, 1 and 2 and the tran-
sition probability matrix
P =

0 1/2 1/2
1/3 0 2/3
0 1 0

(a) Draw the transition diagram for this chain, and show that the chain is
regular.
(b) Find the stationary distribution of the chain.
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B10: (12 marks) Consider a digital communication system that transmits the
digits 0 and 1 through several stages. At each stage the probability that a
0 is received as a 1 is 0.25 and the probability that a 1 is received as a 0 is
0.3. Let Xn = 0 if a 0 entered at the n stage and Xn = 1 if a 1 entered at
the n stage. Then the sequence {Xn : n = 0, 1, 2, . . .} is a Markov chain.
a) Construct the transition probability matrix P .
b) What is the probability that a 0 that is entered at the first stage is
received as a 0 by the fifth stage?
c) Obtain the stationary probabilities pi0, and pi1.
d) Let fnii be the probability that the first return to state i occurs n trans-
missions after leaving i. That is, fnii = P (Xn = i,Xk ̸= i for k =
1, 2, 3, ..., n− 1|X0 = i). Compute f 100, f 200, f 300, fn00 for n ≥ 3.
e) Calculate the mean recurrence time µ0 =
∞∑
n=1
nfn00.
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This page is for answers for Part B questions which you could not fit in the
space provided. Indicate these clearly. Rough work for Part B questions
(not to be graded) should also be done in the Scratch Papers.
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This page is for answers for Part B questions which you could not fit in the
space provided. Indicate these clearly. Rough work for Part B questions
(not to be graded) should also be done in the Scratch Papers.
19
Stats 2503B Final Exam Take Home v111 2020
This page is for answers for Part B questions which you could not fit in the
space provided. Indicate these clearly. Rough work for Part B questions
(not to be graded) should also be done in the Scratch Papers.
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