please answer all parts from a-e below the first image to help you and the second image is the question, please the codes must be clear and I can run it in R so do not send an image or screenshot.

(a) For large n, let U1. .... U. be i.i.d. uniform (0,1) variables. For each i, let X, = sin(U, )e f. Take I = (X, + . . . + X,.). (b) Let E1. .. . . En be i.i.d. exponential variables with parameter ) = 1. Let X, = sin( E.)e-Fie5.. Take I = (X1 + . .. + X,). (c) Let Ni. .... N,, be i.i.d. normal variables with p = 0 and o' = 1/2. Let X, = VTlog(N?). Take I = ( Xit .. . + X,,). (d) Let Gi..... G., be i.i.d. geometric variables with parameter p = 1/2. Let X, = sin(G, ). and take S = (Xi + . .. + X,). (e) Let Pi..... P., be i.id. Poisson variables with parameter A = 1. Let X, = cos(cos Pile. and take S= (X1 4 . .. + X,, ). TaIn 10,000 rolls of two dice there is at least a 95% chance that the sum will be between what two numbers? 9.9 Show how to use Monte Carlo techniques to approximate the following integrals and sums. (a) 1 = [ sin(x)e- dr. (b) 1 = [ sin(x)e- dr. (c) 1 = [ log(23 )e-3 dr. sink (d) 2k 394 LIMITS COS Cos AT- (e)