How to sample from unknown distribution

Question

Let's just assume on the picture we have a prob. distribution of aliens height on a Mars. We know for sure, the area below the curve is 1.

But I have a different question. How do we approach sampling from unique distributions as this is? Taking mean, variance, standard deviation out of it, etc, would not be really helpful, I guess, or it would be? Basically, I am just interested in how to sample from distributions, which are not well defined such as normal distribution etc.

Answer 1

You apply the mathematical inverse of the cumulative distribution function to numbers randomly sampled from a uniform distribution on the interval $[0,1]$.

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Suppose for example you want to sample numbers from the exponential distribution which has a probability density function,

$$ f_X(x) = \frac{1}{\tau} e^{-x/\tau}\qquad (0\leq x ),$$

the cumulative distribution function is defined as,

$$ F(z) = P( X < x)$$ $$= \int_0^z f_X(x) dx $$ $$= \frac{1}{\tau} \int_0^z e^{-x/\tau} dx $$ $$= e^{-z/\tau} - 1 $$

Now we have that,

$$F(z) = 1 - e^{-z/\tau},$$

the mathematical inverse of this function is,

$$F^{-1}(z) =-\tau \log(1-z).$$

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Now we will apply the method I described at the beginning of this answer to get numbers sampled from the exponential distribution.

First I need a source of uniformly random numbers on the interval from $[0,1]$. I will use random.org to generate these numbers.

https://www.random.org/decimal-fractions/

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I generated 100 random number sampled from the uniform distribution on $[0,1]$ using the random.org link above. The histrogram from these numbers follows.

Then I applied $F^{-1}(z)$ to each of these numbers (I chose $\tau=1$). The resulting list of numbers obtained from this process obeys an exponential distribution. Their histogram is shown below.

You can see that the histogram has changed to have a shape consistent with an exponential distribution.