Question

Prove that Box-Muller method described in class generates independent standard normal random variables.

4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample of size n from the geometric distribution with specified success probability p implementing your algorithm in R.

Answer 1

R Programs

#a)

f=function(n,nu,sigma)
{
for(i in 1:n)
{
u=runif(1,0,1)
v=runif(1,0,1)
x=sqrt(-2*log(u))*sin(2*pi*v)
y=sqrt(-2*log(u))*cos(2*pi*v)
print(x)
}
}

#b)

f=function(n,p)
{
lambda=-log(1-p)
for(i in 1:n)
{
u=rexp(1,rate=lambda)
x=trunc(u)
print(x)
}
}