Consider the definite integral

=01exp[xcos(x)]dx

(a) Construct,implement an independent Monte Carlo sampling plan using the uniform density () on [0, 1] and n = 10⁶ independent samples for estimating . Report n, its standard error, and its relative error.

(b) Repeat part (a) using an alternative density function w(-) on [0, 1] that increases compu tational efciency. Keep in mind that variance reduction is likely if wot) > x) where h2(x) = exp[2x cos(:rrx)] is large; so a plot of the latter two functions and your candidate w(-) should help. I suggest that you focus on beta densities wtx) = Mx\"\"'(1 x)'1, x 6 (0,1), NOON!\" where or. 13 :a 0 and the Na) 5 fo 8"!\"\"1 Jr is the gamma function. Since for positive integers a, l"(a) = (a l)!, I suggest focusing on cases where one of the parameters a and is a positive real and the other is equal to 1. In those cases the respective c.d.f. can be inverted easily. Your write-up should include the importance-sampling density and the sampling algorithm from it