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Suppose I want to generate a histogram of $n$ trials a slightly complicated random variable in Mathematica. Is there an easy way to do this?

For instance, to make a histogram of a normal variable one can simple use 

Histogram[RandomVariate[NormalDistribution[0, 1], 200]]

I want to create a histogram of a more complicated random variable. The random variable comes from random matrix theory, but its specific form is probably not so important. If we call it X, running

Histogram[RandomVariate[X, 200]]

returns

RandomVariate::udist: "The specification X is not a random distribution recognized by the system." Indeed, even running

Histogram[RandomVariate[NormalDistribution[0, 1]^2, 200]]

returns this error message, and I think if I understood what's going wrong in this latter case my problem would be resolved also.

There is a Mathematica help page on the topic, but all the examples of random variables given there are for prepackaged probability distributions.

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  • $\begingroup$ Have you looked at ProbabilityDistribution[] $\endgroup$
    – george2079
    Nov 26, 2013 at 21:07
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    $\begingroup$ What do you mean by NormalDistribution[0, 1]^2? It is square of the probability density function or x^2 where x distributed as NormalDistribution[0, 1]? TransformedDistribution, EmpiricalDistribution and SmoothKernelDistribution can be helpful. Could you provide an example of the distribution? $\endgroup$
    – ybeltukov
    Nov 26, 2013 at 21:09
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    $\begingroup$ Break your problem down to pieces and find the smallest piece that you can't get working. Do you need to plot the PDF or do you just need to generate a large number of random variates? It seems to me that you need the latter. In this case Histogram is irrelevant to the question. NormalDistribution[0, 1]^2 is not correct syntax. As george said, look at ProbabilityDistribution. $\endgroup$
    – Szabolcs
    Nov 26, 2013 at 21:09
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    $\begingroup$ You don't have a problem with Histogram, you have a problem defining a distribution. Have a look at Create Your Own Distribution Workshop. $\endgroup$
    – Sjoerd C. de Vries
    Nov 26, 2013 at 21:19

1 Answer 1

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I think this is what you are after..

Histogram[
  RandomVariate[
    ProbabilityDistribution[
     2 Sqrt[π] (PDF[NormalDistribution[0, 1]][x])^2, {x, -∞, ∞}], 2000]]

Note that the argument to ProbabilityDistribution[] has to be a proper probability density function in the sense that it integrates to unity. (Thats where that 2 Sqrt[π] comes from)

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  • $\begingroup$ Thanks! Sorry for the delayed response. This does indeed produce a histogram of X^2 where X is a standard normal variable. I'm afraid I oversimplified the problem in this example though. How would one go about creating a histogram, summing a series of independent identically distributed variables for instance? The actual problem I want to solve concerns operations on the eigenvalues of random matrices. Would it be most appropriate to simply create a new question, or to edit the question above to be more precise? $\endgroup$
    – Brad Rodgers
    Nov 29, 2013 at 15:46

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