Generating Geometric and Truncated Exponential Random Variates

If you do numerical simulations you often find yourself needing to generate random variates from a specific distribution.  There are several techniques for doing this, such as the inverse transform method and the acceptance-and-rejection method.  This post will talk about a simple way to generate variates from the geometric and truncated exponential distributions, where the latter is the exponential distribution restricted to the range [0,1].

The classic example of the inverse transform method is generating random variates from an exponential(\lambda) distribution.  (This is because the exponential cdf is so easy to invert.)  If U \sim uniform(0,1), and X = -\ln U/\lambda, then X \sim exponential(\lambda).  So it’s easy to generate exponential random variates.

Then, to get geometric and truncated exponential random variates, all you have to do is take \lfloor X \rfloor and \{X\}, the integer and fractional parts of X, respectively!  More explicitly, the integer part \lfloor X \rfloor has a geometric(1-e^{-\lambda}) distribution, and the fractional part \{X\} has the truncated exponential(\lambda) distribution on (0,1).

Proof:

\displaystyle P(\lfloor X \rfloor = k, \{X\} \leq x) = P(k \leq X < k + x) = e^{-\lambda k} - e^{-\lambda (k+x)} = e^{-\lambda k} (1 - e^{-\lambda x}).

If we fix k and substitute 1 for x we have P(\lfloor X \rfloor = k, \{X\} \leq 1) = P(\lfloor X \rfloor = k) = e^{-\lambda k} (1 - e^{-\lambda}) = (e^{-\lambda})^k (1 - e^{-\lambda}).
Thus \lfloor X \rfloor \sim geometric(1-e^{-\lambda}).

If we fix x and sum over all possible values of k we have \displaystyle P(\{X\} \leq x) = (1 - e^{-\lambda x}) \sum_{k=0}^{\infty} e^{-\lambda k} = \frac{1 - e^{-\lambda x}}{1- e^{-\lambda}}.

Now, f(x) = 1 - e^{-\lambda x} is the cdf of an exponential(\lambda) distribution, and 1- e^{-\lambda} is the probability that an exponential(\lambda) random variable is no greater than 1.  Thus \{X\} has this truncated exponential(\lambda) distribution — an exponential(\lambda) distribution whose values are not allowed to be greater than 1.

See, for example, Eisenberg [1] or my answer to “Random exponential-like distribution” on math.SE.

References

1.  Bennett Eisenberg, “On the expectation of the maximum of IID geometric random variables,” Statistics and Probability Letters 78 (2008) 135-143.

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