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MTH 511A : Statistical Simulation and Data Analysis-2017
Assignment 1
1.
If x0  5 and xn  3xn1 mod 150, find x1 ,.., x10 .
2.
If x0  3 and xn  (5 xn1  7) mod 200, find x1 ,.., x10 .
3.
Approximate the following integrals using simulation and compare the estimates with the exact
answer, wherever possible.
1
(i)

exp e x  dx
1
(ii)
0
2 3/ 2
 (1  x ) dx
x
 e dx
2
1 1
(vi)

e
0 0
( x  y )2
dydx
x x
 e dx
2
2
 x
0

(v)

2
(iii)
(vii)
e
( x y )
(iv)
 x(1  x
2 2
) dx
0
dydx
0 0
1 if y  x
[Hint: Let I y ( x)  
and use this function and use this function to equate the integral
 0 if y  x
to one in which both terms go from 0 to  .]
4.
Use simulation to approximate Cov (U , eU ), where U is uniform on (0, 1). Compare your
approximation with the exact answer.
5.
Let U be uniform on (0, 1). Use simulation to approximate the following:
(a) Correlation (U , 1  U 2 ).
(b) Correlation( U 2 , 1  U 2 ).
6.
 n

For uniform (0, 1) random variables U1 ,U 2 ,... , define N=Minimum n :  U i  1 , that is, N is
 i 1

equal to the number of random numbers that must be summed to exceed 1.
(a) Estimate E[N] by generating 100 values of N.
(b) Estimate E[N] by generating 1000 values of N.
(c) Estimate E[N] by generating 10,000 values of N.
(d) What do you think is the value of E[N].
7.
xn  3xn1  5xn2 mod(100), n  3 We will call the sequence
With x1  23, x2  66, and
un  xn /100, n  1, the text’s random number sequence. Find its first 14 values.