Simulating a non-simple exponential distribution

Question

I want to simulate a vector x in R with 1000 entries. The j'th entry comes from a exponential distribution with density

$$ f_{j}(x)=j \beta e^{-j \beta x}, x>0 $$

Let us just say that $\beta=2$.

So the first entry in x comes from an exponential distribution with density $$f_{1}(x)=1 \beta e^{-1 \beta x}, x>0$$

The second entry in x comes from an exponential distribution with density $$f_{2}(x)=2 \beta e^{-2 \beta x}, x>0$$

etc.

Answer 1

R will happily vectorize the rate parameter:

beta <- 2
rexp(1000,rate=(1:1000)*beta)

Answer 2

General method (shown in R) that does not rely on how R's rexp works.

set.seed(2021)
n = 1:1000
x = -(1:1000)*log(runif(1000))

plot(n,x)