Use this function to equate the integral to one in which both terms go

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Statistical Computing

Notes about your assignment:  You should use R to answer each of these questions.  In your R script please label the questions in order using comments (as well as any sub-parts), and include the answer to each question (typically, what you get when you run your code) as comments at the end of each question.

1. If x0 = 5 and

find x1, ..., x10

2. If x0 = 3 and 

xn = 3xn−1 mod 150

find x1, ..., x10 

xn = (5xn−1 + 7) mod 200

For questions 3-9, use Monte Carlo integration to approximate the given integrals. If you are able to calculate the integral analytically, you may want to use this to double-check your work!

3. ∫ 1 eex dx

4. ∫ 1(1 − 2 3

0

5. 2

−2 

x ) 2 dx ex+x2 dx 

6. ∫ ∞ x(1 + x2)−2dx 

7. ∞  ex2 dx

−∞

[Hint: symmetry is your friend!] 

8. ∫ 1 ∫ 1 e(x+y)2 dydx

9. ∫ ∞ ∫ x e−(x+y)dydx

[Hint: Let IY 

(x) = 1  y < x . Use this function to equate the integral to one in which both terms go

0  y ≥ x from 0 to ∞]

10. Use simulation to approximate Cov(U, eU ) where U is uniform on (0, 1).

11. Let U be uniform on (0, 1). Use simulation to approximate the following:

a) Cov(U, √1 − U 2)

b) Cov(U 2, √1 − U 2)

12. For uniform (0, 1) random variables U1, U2, ... define N to be the number of random numbers that must be summed to exceed 1:

N = Minimum{n : Σn Ui > 1}

a) Estimate E{N} by generating 100 values of N 

b) Estimate E{N} by generating 1,000 values of N

c) Estimate E{N} by generating 10,000 values of N

d) Can you guess the true value of E{N}?

13. For uniform (0, 1) random variables, define N to be the maximum number of random numbers whose product is still at least e−3:

N = Maximum{n : Qn Ui ≥ e−3}

a) Find E{N} by simulation

b) Find P (N = i) for i = 0, 1, 2, 3, 4, 5, 6 by simulation

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