How to sample from an exponential distribution using rejection sampling in PHP

Question

I wrote PHP-code that gets me samples of a (truncated) exponential distribution between 0 and 1 with mean 1 ($X\sim \mathrm{Exp}(1)$).

I'm trying to use acceptance-rejection method: I don't know if I get it right, but basically I use one sample of a $\mathrm{Uniform}(0,1)$ distribution to get the x-axis and another to get the y-axis and then check if the y-axis sample is below (valid sample) or above (reject sample) the exponential distribution function curve.

How do I get it to give me samples of a generic $X\sim \mathrm{Exp}(\lambda)$?

$num_samples=1000;
$samples=array();$sample=0;$count=0;$counts=0;
$time=microtime(true);
for($i=0;$i<$num_samples;$i++){
    while (exp(-($sample = mt_rand(0,PHP_INT_MAX)*(1/PHP_INT_MAX)))
                <
                mt_rand(0,PHP_INT_MAX)*(1/PHP_INT_MAX)){
        $count++;
    }
    $samples[] = $sample;
    $counts+=$count;
    $count=0;
}
echo "samples: ".$num_samples." time: ".(microtime(true)-$time)." efficiency: ".$num_samples/$counts."\n";

Answer 1

Your exponential random variable $X$ truncated between $0$ and $\tau>0$ has distribution function $$ F_X(x) = \begin{cases} 0 & \textrm{if} & x\leq 0 \,;\\ \frac{1-e^{-\lambda x}}{1-e^{-\lambda\tau}} & \textrm{if} & 0<x<\tau \, ;\\ 1 & \textrm{if} & x \geq \tau \, .\\ \end{cases} $$

Using this result, we can generate realizations of $X$ with the following PHP code.

<?php 
    $lambda = 1;
    $tau = 10;
    $u = mt_rand(0, PHP_INT_MAX) / PHP_INT_MAX;
    $x = - log(1 - (1 - exp(- $lambda * $tau)) * $u) / $lambda;
    echo $x;
?> 

samples: 1000000 time: 3.1613190174103 efficiency: 1

Answer 2

Ok, I just had a moment to take another look at it and following @Glen_b directions I could figure it out. It's terribly inefficient, but in case anybody is interested, here is the code:

$num_samples=1000;
$avg = 2; /* ($lambda = 1/$avg) */
$trunc = 10; /* ($tau = $trunc*$avg) */
$start_time = microtime(true);
for($i=0;$i<$num_samples;$i++){
do {
    $sample_exp_lambda_1_tau_10 = mt_rand(0,PHP_INT_MAX)*($trunc/PHP_INT_MAX);
    $p_exp_lambda_1_tau_10 = exp(-$sample_exp_lambda_1_tau_10);
    $y_axis = mt_rand(0,PHP_INT_MAX)*(1/PHP_INT_MAX);
    $count++;
} while ($p_exp_lambda_1_tau_10 < $y_axis);
$samples[] = $avg*$sample_exp_lambda_1_tau_10; /* ($x =  $avg*$sample_exp_lambda_1_tau_10) */
$total_count+=$count;
$count=0;
}
echo "samples: ".$num_samples." time: ".(microtime(true)-$start_time)." efficiency: ".$num_samples/$total_count."\n";

And that's all. With this input I generated 1000 samples of a X~exp(2) truncated at 20.

Here is the output:

samples: 1000 time: 0.018975019454956 efficiency: 0.1012555690563

And here the histogram:

(Close enough.)