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MTH 511A : Statistical Simulation and Data Analysis-2017 Assignment 1 1. If x0 5 and xn 3xn1 mod 150, find x1 ,.., x10 . 2. If x0 3 and xn (5 xn1 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 3xn1 5xn2 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.