The goal of this problem is to simulate a geometric random variable (which will be covered on Friday, Feb 6th in lecture). We will break this problem into two parts.

Part a

Build a function which takes in p. This function should repeat a Bernoulli trial until a "success" (defaulted to a 1 in R) occurs, and return the trial number on which the first success occurred (i.e., if the 3rd trial resulted in a "success", this function should return 3). Use your function with the following values of p (and show your functions output):

i.p=0.02 ii.p=0.10 iii. p = 0.50

Part b

Build another function which takes in p, r, and N. This function will calculate the probability that the first "success" occurred on the rth trial, based on N simulations.

Notice you can use your function from Part a in this function.

Use your function with the following values of p, r, and N (and show your functions output):

i. p = 0.02, r = 10, N = 600000
ii. p = 0.10, r = 10, N = 600000
iii. p = 0.20, r = 10, N = 600000

These simulations may take a few minutes (or more depending on your computer) to run, but they should not crash your computer.

Remember that you can always use the computer labs to complete simulations.