Problem 1 "Monte Carlo integration" Consider the following integrals: A = exp(-1) Vidz 1 = / exp(-2)Vidz c= [ VI+ cos(3 )de D= VI+ sin(3Jar For checking your code, the values of the integrals are A =: 0.8862269255, # # 0.8860764953, C = 2.712040395 and D = 1.917702155. a) Write down an expression for a Monte Carlo estimator of A based on a random draws from an Exponential (1) distribution. b) Write down an expression for a Monte Carlo estimator of # based on a random draws from an Exponential (1) distribution. ") Write down an expressions for Monte Carlo estimators of C and D based on n random draws from a Uniform(-1, 1) distribution. i) Write down an expressions for importance sampling estimators of A and # based on n random draws from a Gamma(3/2, 1) distribution. Do you, based on theory, expect these to work well? e) Write down an expressions for importance sampling estimators of A and # based on n random draws from a Gamma(4, 1/4) distribution (i.e. with scale parameter equal to 1/4). Do you, based on theory, expect these to work well? f) Write down an expressions for antithetic Monte Carlo estimators of ( and D based on random draws from a Uniform(-1, 1) distribution and a evaluations of the integrands. Do you, based on theory, expect these to work well? g)"Implement the estimators you proposed in points d), e) and f), and verify that your theoretical predictions concerning the behavior of the estimators are correct. 2