Comment by pash

1 year ago

The general version of this is called inverse transform sampling [0], which uses the fact that for the cdf F of any random variable X the random variable Y = F(X) has a standard uniform distribution [1]. Since every cdf increases monotonically on the unit interval, every cdf is invertible [2]. So apply the inverse cdf to both sides of the previous equation and you get F^-1(Y) = X is distributed like X.

Sampling from a standard uniform distribution and then using the inverse transform is the commonest way of generating random numbers from an arbitrary distribution.

0. https://en.m.wikipedia.org/wiki/Inverse_transform_sampling

1. https://en.m.wikipedia.org/wiki/Probability_integral_transfo...

2. Not every cdf is one-to-one, however, so you may need a generalized inverse.

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pash

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tmoertel  1 year ago

For the particular case of the exponential distribution we can go further. By taking advantage of the theory of Poisson processes, we can take samples using a parallel algorithm. It even has a surprisingly succinct SQL translation:

    SELECT *
    FROM Population
    WHERE weight > 0
    ORDER BY -LN(1.0 - RANDOM()) / weight
    LIMIT 100  -- Sample size.

Notice our exponentially distributed random variable on prominent display in the ORDER BY clause.

If you're curious, I explore this algorithm and the theory behind it in https://blog.moertel.com/posts/2024-08-23-sampling-with-sql....

  • lkuty  1 year ago

    Quite off-topic, but do you know when you'll write the article about CPS, if ever?

    • tmoertel  1 year ago

      Oops. I had quite forgotten that I need to write about that. I said I would over a decade ago, so that's a long time for you to wait. Sorry about that.

      I mainly write for myself, so I need the time and the motivation. Until recently, my job at G took up my time and also provided an internal community where I could scratch the writing itch, which reduced the motivation for public writing on my blog. But now that I'm semi-retired, I'll try to write more frequently.

      Thanks for the accountability!