How to sample N points between 0 and R if they are exponentially distributed?

Question

The density of my points x ∈ [0,R] is exponential: ρ(x)~e^x How can I sample N points from there?

Answer 1

Taking your request at face value, if you want a density function that grows exponentially for x ∈ [0,R] the cumulative distribution function turns out to be (exp(x) - 1) / (exp(R) - 1). To generate this via inversion, set the CDF equal to a Uniform(0,1) and solve for x. The inversion turns out to be:

ln(1 + (exp(R) - 1) * U)

where U represents a call to the Uniform(0,1) PRNG.

If what you actually want is a truncated form of what most probability folks know as the exponential distribution, we need to determine an upper bound for the random number corresponding to your truncation point R. In that case, the inversion is:

-ln(1 - [1 - exp(-lambda * R)] * U) / lambda

As before, U represents a call to the Uniform(0,1) PRNG. This will generate exponentials at rate lambda, truncated at a max of R.

Answer 2

Use inverse sampling: you generate uniform distributed values and map them to the output of the cdf of your distribution.