1. Program up our "alternative" algorithm for sim...

1. Program up our "alternative" algorithm for simulating a Poisson TV X with mean a. at P(X = k) = e Here is the Pseudo Code: Alternative algorithm for generating a Poisson rv x with mean a: i Enter desired a > 0. Set X = 0, P = 1 ii Generate U ~ unif(0,1), set P = UP iii If P e then stop. Otherwise if P209 then set x = x+1 and go back to ix. iii Output X. (a) For o = 1, and o = 5. run the algorithm 1000 times to obtain n = 1000 copies of X. and see how close your empirical average is to the true value E(X) = 1. E(X) 5: 1000 1 E(X) 2 1000 [x Re-do for 72 = 10,000 copies of X. (b) For a = 5, P(X = 5) = €-556 = 0.17546.. See how close the following simulation estimate is to the correct answer, for n = 1000 and n = 10,000 (generated copies of X): 2 n where 1 {A} is the indicator rv for the event A: 1(A} = 1 if Aoccurs, 1(A} = 0 if A des not occur. 2. Recall the famous irrational number IT = 3.1415926535. We know that it can be expressed as the area of a disk of radius r = 1: A = xr2 = II. and hence a = 4 = where U is a continuous uniform TY over (0,1) (it has density function f(x) = 1, x € (0,1).) Obtain an approximation to * by generating 1000 iid U, and using 1000 1 x 2 4 1000 1 NOTE: When obtaining an empirical average estimate of the form E(X) i-1 (n some large integer) for efficiency you do not want to first generate and save all n of the X,. Instead you want to sum them up sequentially thus only saving one at a time: Initialize: SUM = 0. Count = 1 1. Generate X. Reset SUM = SUM+X. 2. If Count 3. Output SUM/n. 2