There is usually more than one way to simulate a particular random variable. In this exercise we consider two ways to generate geometric random variables. a. We give you a sequence of independent U(0,...

There is usually more than one way to simulate a particular random variable. In this exercise we consider two ways to generate geometric random variables.

a. We give you a sequence of independent U(0, 1) random variables U₁, U₂, . . . . From this sequence, construct a sequence of Bernoulli random variables. From the sequence of Bernoulli random variables, construct a (single) Geo(p) random variable

b. It is possible to generate a Geo(p) random variable using just one U(0, 1) random variable. If calls to the random number generator take a lot of CPU time, this would lead to faster simulation programs. Set λ = − ln(1− p) and let Y have a Exp(λ) distribution. We obtain Z from Y by rounding to the nearest integer greater than Y . Note that Z is a discrete random variable, whereas Y is a continuous one. Show that, nevertheless, the event {Z>n} is the same as {Y >n}. Use this to compute P(Z>n) from the distribution of Y . What is the distribution of Z? (See Quick exercise 4.6.)