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程序代写案例-2503B

时间：2021-04-27

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.

Stats 2503B Final Exam Take Home v111 2020

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

2

Stats 2503B Final Exam Take Home v111 2020

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

3

Stats 2503B Final Exam Take Home v111 2020

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

4

Stats 2503B Final Exam Take Home v111 2020

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

5

Stats 2503B Final Exam Take Home v111 2020

B1 (continued)

6

Stats 2503B Final Exam Take Home v111 2020

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.

7

Stats 2503B Final Exam Take Home v111 2020

B3: (6 marks) Use the method of variation of parameters to obtain a particular

solution of

y′′ + 4y = sec 2t

8

Stats 2503B Final Exam Take Home v111 2020

B4: (6 marks) Find L−1

{

1

(s2 + 4)2

}

.

9

Stats 2503B Final Exam Take Home v111 2020

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)

.

10

Stats 2503B Final Exam Take Home v111 2020

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).

11

Stats 2503B Final Exam Take Home v111 2020

B6 (continued)

12

Stats 2503B Final Exam Take Home v111 2020

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).

13

Stats 2503B Final Exam Take Home v111 2020

B7 (continued)

14

Stats 2503B Final Exam Take Home v111 2020

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).

15

Stats 2503B Final Exam Take Home v111 2020

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.

16

Stats 2503B Final Exam Take Home v111 2020

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.

17

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.

18

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.

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.

20

学霸联盟

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.

Stats 2503B Final Exam Take Home v111 2020

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

2

Stats 2503B Final Exam Take Home v111 2020

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

3

Stats 2503B Final Exam Take Home v111 2020

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

4

Stats 2503B Final Exam Take Home v111 2020

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

5

Stats 2503B Final Exam Take Home v111 2020

B1 (continued)

6

Stats 2503B Final Exam Take Home v111 2020

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.

7

Stats 2503B Final Exam Take Home v111 2020

B3: (6 marks) Use the method of variation of parameters to obtain a particular

solution of

y′′ + 4y = sec 2t

8

Stats 2503B Final Exam Take Home v111 2020

B4: (6 marks) Find L−1

{

1

(s2 + 4)2

}

.

9

Stats 2503B Final Exam Take Home v111 2020

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)

.

10

Stats 2503B Final Exam Take Home v111 2020

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).

11

Stats 2503B Final Exam Take Home v111 2020

B6 (continued)

12

Stats 2503B Final Exam Take Home v111 2020

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).

13

Stats 2503B Final Exam Take Home v111 2020

B7 (continued)

14

Stats 2503B Final Exam Take Home v111 2020

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).

15

Stats 2503B Final Exam Take Home v111 2020

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.

16

Stats 2503B Final Exam Take Home v111 2020

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.

17

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.

18

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.

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.

20

学霸联盟