How can I show transformation from uniform distribution to exponential distribution in R?

Question

When variable X has uniform distribution (0,1), I can make a formula Y = -(lambda)lnX and the distribution of Y would show Exp(lambda).

And I'm trying to prove this case when lambda is 3, showing that two distribution curves match. I've made it this far, but can't figure out really how.

w <- seq(0, 10, length=500)
x <- dunif(w, 0, 1)
y <- (-1)*(3)*log(x)
z <- dexp(w, 3)
plot(w, y, type="l")
par(new=F)
plot(w, z)

Answer 1

This is a problem related to inverse CDF method.

First of all, you get inverse CDF wrong. It is -1/3, not -3.

Secondly, as the name suggests, it is inverse CDF, not inverse PDF. You can't do such transformation on density function. Instead, draw samples and use quantile-quantile plot.

n <- 500
x <- runif(n)
y <- -1/3 * log(x)
z <- rexp(n, 3)
qqplot(y, z)
abline(0, 1)

Alternatively, compare empirical CDF with theoretical CDF.

plot(ecdf(y))
curve(pexp(x, 3), add = TRUE, col = 2)