xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

R代写-729G/4

时间：2021-04-22

Stat-461/Mast-729G/4 (2020-21)

Sample questions for final exam

1. Suppose that a store opens at 0pm and customers arrive at the store according

to a non-homogeneous Poisson process {N(t), 0 ≤ t ≤ 8} with the intensity

function

λ(t) =

{

4− t

4

, 0 ≤ t ≤ 4,

3 + t−4

4

, 4 < t ≤ 8.

Let Sj denote the time at which the j-th customer arrives, j ≥ 1. Compute

(a) P [S11 ≤ 5|N(3.5) = 9].

(b) P [S2 ≤ 4|N(6) = 4].

2. Suppose N(t) is a Poisson process with rate function λ(t) = | sin(t)|, t ≥ 0.

Denote the event-times of N(·) by 0 < S1 < S2 < · · · . Find P [S1 ≤ pi < S3],

and generate the first two event-times S1, S2.

3. Let

N = min

{

n ≥ 1 :

∣∣∣∣U21 + · · ·+ U2nn − 0.333

∣∣∣∣ < 0.001} ,

where U1, U2, . . . are iid Uniform (0, 1) random variables. Estimate E(1/N)

by k−1

∑k

i=1(1/Ni) where k ≥ 200 and sd(1/N)/sqrt(k)< 0.004.

4. For independent sequences of iid Uniform (0, 1) random variables {U1, U2, . . .}, {V1, V2, . . .}

define

N = min

{

n : Un ≤ 1

Vn + 1

}

.

Obtain E(N) and joint as well as marginal pdf’s of (UN , VN).

5. (a) Suppose

P (X = i) = 0.2 for i = 1, . . . , 5.

We want to generate random numbers from X by the acceptance-rejection

method using

P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .

for some 0 < p < 1. Find p that minimizes(

max

i

P (X = i)

P (Y = i)

)

and generate a value of X using that p.

1

(b) Suppose

P (X = i) =

i

15

for i = 1, . . . , 5.

We want to generate random numbers from X by the acceptance-rejection

method using

P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .

for some 0 < p < 1. Find p that minimizes(

max

i

P (X = i)

P (Y = i)

)

and generate a value of X using that p.

6. Let X ∼ Geometric (1/2) and Y ∼ Geometric (1/3) be two independent

random variables. Obtain the conditional distribution of X, given that X =

Y +1, and then generate a value of a random variable having this conditional

distribution.

7. Generate a value of a random variable X with p.d.f.

f(x) = xe−x

2

+

1

2

e−x, x ≥ 0,

using the indicated methods:

(a) Use the acceptance/rejection method, with g(x) = e−x, x ≥ 0, distribu-

tion as g.

(b) Use the composition method.

8. Generate a value of a random vector (X1, X2) with joint pdf

f(x1, x2) = 3(x1 + x2), 0 < x1, 0 < x2, x1 + x2 ≤ 1,

by acceptance-rejection method, using another random vector (Y1, Y2) with

joint pdf

g(y1, y2) =

1

4

√

y1y2

, 0 < yi ≤ 1, i = 1, 2.

What is the expected number of rejections?

9. Suppose (X1, X2) have the joint pdf f(x1, x2) = 2(x1+x2)I(0 ≤ x1 < x2 ≤ 1).

Find the marginal pdf f2(x2) of X2, the conditional pdf f(x1|x2) of X1, given

X2 = x2, and hence, or otherwise, generate a value of (X1, X2).

2

10. Consider a single-server queueing model where customers arrive according to

a homogeneous Poisson process with rate λA = 10. Upon arrival a customer

either enters service if the server is free at that moment or else joins the waiting

queue if the server is busy. When the server completes serving a customer it

then begins serving the customer that had been waiting the longest. If there are

no waiting customers the server remains idle until the next customer arrives.

The service times are i.i.d. Exponential random variables with rate λS = 12,

independent of arrivals.

(a) Generate the arrival times of the first three customers and their departure

times.

(b) Generate the arrival times of the first three customers and their departure

times, given that both Customer-2 and Customer-3 have tolerance times

which are i.i.d. Exponential (15) random variables, i.e., each will leave

after an Exponential (15) amount of time from his/her arrival, if not

served by then.

11. Consider a queue with 2 parallel servers, where server i takes Exponential (i)

amount of time to complete a service, i = 1, 2. Services are independent of one

another and of the arrivals which occur according to a homogeneous Poisson

process with rate 2. Every arrival goes to the idle server if one of the servers

is busy, to server 1 if both servers are free and wait in a common queue if both

servers are busy; the one that has been waiting the longest is served first when

a server becomes idle. Simulate the arrival and departure times of the first 3

arrivals, and calculate the idle times of the two servers until the departure of

the 3rd arrival.

12. Suppose that X has the pmf:

P (X = x) = 3

∫ 1

0

y(1− y)x+1dy, x = 1, 2 . . . .

Generate a value of X by the composition method and also a value of an

antithetic variable X ′ for X.

13. Suppose that (X,Y ) have the joint p.d.f.

f(x, y) =

{

2

√

x, if 0 ≤ x ≤ 1 and 0 < y < √x,

0, otherwise.

We want to use an estimator of the form [Z + c(X − E(X))] to estimate

P{X ≤ 2Y }, where

Z =

{

1, if X ≤ 2Y,

0, otherwise.

3

Find the (theoretical) optimal choice of c.

14. Suppose that (X,Y ) have the joint p.d.f.

f(x, y) = 4e−

x

y

−2y2 , x > 0, y > 0.

We want to estimate θ = P{|X − Y | < 1}. Consider

Z =

{

1, if |X − Y | < 1,

0, otherwise.

(a) Obtain g(Y ) = E(Z|Y ).

(b) Compute

∑2

i=1 g(Yi)/2 by generating Y1 and Y2.

15. Let X = eU , where U is a Uniform (0, 1) random variable.

(a) Compute Var((X +X ′)/2), where X ′ is antithetic to X.

(b) Obtain c that minimizes Var(X + c(U − 0.5)).

16. Suppose that Y is an Exponential (1/2) random variable, and conditional on

Y = y, N is a Poisson random variable with mean y. We want to estimate

p = P{2 ≤ N ≤ 3}. Obtain E(Z|Y ), where

Z =

{

1, if 2 ≤ N ≤ 3,

0, otherwise.

Compute the estimate

∑5

i=1E(Zi|Yi)/5 by generating 5 i.i.d. copies Y1, . . . , Y5

of Y .

Values of Uniform (0, 1) Random Variables

0.81222166 0.05464751 0.06930601 0.94253943 0.79179519

0.49573802 0.96771072 0.89496062 0.89866417 0.85499261

0.67242223 0.33383212 0.08998713 0.67070207 0.32724888

0.24365016 0.14752814 0.57458281 0.82379935 0.48721456

0.37257331 0.22673586 0.22982463 0.96236055 0.87142464

0.64248448 0.15947970 0.31127181 0.37530767 0.46249647

0.22759288 0.61511912 0.42639104 0.00558105 0.57204456

0.78462924 0.83509543 0.67738296 0.99870467 0.29117284

4

学霸联盟

Sample questions for final exam

1. Suppose that a store opens at 0pm and customers arrive at the store according

to a non-homogeneous Poisson process {N(t), 0 ≤ t ≤ 8} with the intensity

function

λ(t) =

{

4− t

4

, 0 ≤ t ≤ 4,

3 + t−4

4

, 4 < t ≤ 8.

Let Sj denote the time at which the j-th customer arrives, j ≥ 1. Compute

(a) P [S11 ≤ 5|N(3.5) = 9].

(b) P [S2 ≤ 4|N(6) = 4].

2. Suppose N(t) is a Poisson process with rate function λ(t) = | sin(t)|, t ≥ 0.

Denote the event-times of N(·) by 0 < S1 < S2 < · · · . Find P [S1 ≤ pi < S3],

and generate the first two event-times S1, S2.

3. Let

N = min

{

n ≥ 1 :

∣∣∣∣U21 + · · ·+ U2nn − 0.333

∣∣∣∣ < 0.001} ,

where U1, U2, . . . are iid Uniform (0, 1) random variables. Estimate E(1/N)

by k−1

∑k

i=1(1/Ni) where k ≥ 200 and sd(1/N)/sqrt(k)< 0.004.

4. For independent sequences of iid Uniform (0, 1) random variables {U1, U2, . . .}, {V1, V2, . . .}

define

N = min

{

n : Un ≤ 1

Vn + 1

}

.

Obtain E(N) and joint as well as marginal pdf’s of (UN , VN).

5. (a) Suppose

P (X = i) = 0.2 for i = 1, . . . , 5.

We want to generate random numbers from X by the acceptance-rejection

method using

P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .

for some 0 < p < 1. Find p that minimizes(

max

i

P (X = i)

P (Y = i)

)

and generate a value of X using that p.

1

(b) Suppose

P (X = i) =

i

15

for i = 1, . . . , 5.

We want to generate random numbers from X by the acceptance-rejection

method using

P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .

for some 0 < p < 1. Find p that minimizes(

max

i

P (X = i)

P (Y = i)

)

and generate a value of X using that p.

6. Let X ∼ Geometric (1/2) and Y ∼ Geometric (1/3) be two independent

random variables. Obtain the conditional distribution of X, given that X =

Y +1, and then generate a value of a random variable having this conditional

distribution.

7. Generate a value of a random variable X with p.d.f.

f(x) = xe−x

2

+

1

2

e−x, x ≥ 0,

using the indicated methods:

(a) Use the acceptance/rejection method, with g(x) = e−x, x ≥ 0, distribu-

tion as g.

(b) Use the composition method.

8. Generate a value of a random vector (X1, X2) with joint pdf

f(x1, x2) = 3(x1 + x2), 0 < x1, 0 < x2, x1 + x2 ≤ 1,

by acceptance-rejection method, using another random vector (Y1, Y2) with

joint pdf

g(y1, y2) =

1

4

√

y1y2

, 0 < yi ≤ 1, i = 1, 2.

What is the expected number of rejections?

9. Suppose (X1, X2) have the joint pdf f(x1, x2) = 2(x1+x2)I(0 ≤ x1 < x2 ≤ 1).

Find the marginal pdf f2(x2) of X2, the conditional pdf f(x1|x2) of X1, given

X2 = x2, and hence, or otherwise, generate a value of (X1, X2).

2

10. Consider a single-server queueing model where customers arrive according to

a homogeneous Poisson process with rate λA = 10. Upon arrival a customer

either enters service if the server is free at that moment or else joins the waiting

queue if the server is busy. When the server completes serving a customer it

then begins serving the customer that had been waiting the longest. If there are

no waiting customers the server remains idle until the next customer arrives.

The service times are i.i.d. Exponential random variables with rate λS = 12,

independent of arrivals.

(a) Generate the arrival times of the first three customers and their departure

times.

(b) Generate the arrival times of the first three customers and their departure

times, given that both Customer-2 and Customer-3 have tolerance times

which are i.i.d. Exponential (15) random variables, i.e., each will leave

after an Exponential (15) amount of time from his/her arrival, if not

served by then.

11. Consider a queue with 2 parallel servers, where server i takes Exponential (i)

amount of time to complete a service, i = 1, 2. Services are independent of one

another and of the arrivals which occur according to a homogeneous Poisson

process with rate 2. Every arrival goes to the idle server if one of the servers

is busy, to server 1 if both servers are free and wait in a common queue if both

servers are busy; the one that has been waiting the longest is served first when

a server becomes idle. Simulate the arrival and departure times of the first 3

arrivals, and calculate the idle times of the two servers until the departure of

the 3rd arrival.

12. Suppose that X has the pmf:

P (X = x) = 3

∫ 1

0

y(1− y)x+1dy, x = 1, 2 . . . .

Generate a value of X by the composition method and also a value of an

antithetic variable X ′ for X.

13. Suppose that (X,Y ) have the joint p.d.f.

f(x, y) =

{

2

√

x, if 0 ≤ x ≤ 1 and 0 < y < √x,

0, otherwise.

We want to use an estimator of the form [Z + c(X − E(X))] to estimate

P{X ≤ 2Y }, where

Z =

{

1, if X ≤ 2Y,

0, otherwise.

3

Find the (theoretical) optimal choice of c.

14. Suppose that (X,Y ) have the joint p.d.f.

f(x, y) = 4e−

x

y

−2y2 , x > 0, y > 0.

We want to estimate θ = P{|X − Y | < 1}. Consider

Z =

{

1, if |X − Y | < 1,

0, otherwise.

(a) Obtain g(Y ) = E(Z|Y ).

(b) Compute

∑2

i=1 g(Yi)/2 by generating Y1 and Y2.

15. Let X = eU , where U is a Uniform (0, 1) random variable.

(a) Compute Var((X +X ′)/2), where X ′ is antithetic to X.

(b) Obtain c that minimizes Var(X + c(U − 0.5)).

16. Suppose that Y is an Exponential (1/2) random variable, and conditional on

Y = y, N is a Poisson random variable with mean y. We want to estimate

p = P{2 ≤ N ≤ 3}. Obtain E(Z|Y ), where

Z =

{

1, if 2 ≤ N ≤ 3,

0, otherwise.

Compute the estimate

∑5

i=1E(Zi|Yi)/5 by generating 5 i.i.d. copies Y1, . . . , Y5

of Y .

Values of Uniform (0, 1) Random Variables

0.81222166 0.05464751 0.06930601 0.94253943 0.79179519

0.49573802 0.96771072 0.89496062 0.89866417 0.85499261

0.67242223 0.33383212 0.08998713 0.67070207 0.32724888

0.24365016 0.14752814 0.57458281 0.82379935 0.48721456

0.37257331 0.22673586 0.22982463 0.96236055 0.87142464

0.64248448 0.15947970 0.31127181 0.37530767 0.46249647

0.22759288 0.61511912 0.42639104 0.00558105 0.57204456

0.78462924 0.83509543 0.67738296 0.99870467 0.29117284

4

学霸联盟