Inversion Sampling

Prop If , then . Proof Thus .

Algorithm Inversion Sampling To draw samples from CDF , we can

1. Draw
2. Set
3. Return as iid samples from . e.g. Assume is an exponential random variable with rate parameter . Recall that the probability density function is for . First, we compute the CDF:Solving for the inverse CDF, we get that Using our algorithm above, we first generate , then set . We do this in the R code below and compare the histogram of our samples with the true density of :

# inverse transform sampling
num.samples <-  1000
U           <-  runif(num.samples)
X           <- -log(1-U)/2
 
# plot
hist(X, freq=F, xlab='X', main='Generating Exponential R.V.')
curve(dexp(x, rate=2) , 0, 3, lwd=2, xlab = "", ylab = "", add = T)