I would like to plot the PDF of the product of a $Beta(1,1)$ and $Beta(3/2,1/2)$ random variables. I can use the following to calculate the PDF at a point $x$:

PDF[TransformedDistribution[u v, {u \[Distributed] BetaDistribution[1, 1], 
v \[Distributed] BetaDistribution[3/2, 1/2]}], x]

However, to work out the PDF at a point (take $x=0.5$ for example), the above takes about 10 minutes on my computer.

I could leave my computer overnight, and it should be able to plot the above PDF using $Plot$, however, I was wondering whether there are any quicker alternatives?

Note: I want to plot the distribution's PDF exactly, not an approximation by drawing randomly first a $Beta(1,1)$ then multiplying it by a draw from $Beta(3/2,1/2)$, then looking at a histogram.




1 Answer 1


From what I can tell you are recalculating the transformed distribution too often in your plot. Calculate and store the resulting PDF once and then use it for your plots.

tpdf = PDF[
   u v, {u \[Distributed] BetaDistribution[1, 1], 
    v \[Distributed] BetaDistribution[3/2, 1/2]}], x]

Once you have tpdf your plot will return immediately since you are not recalculating it for each value of x.

Plot[tpdf, {x, 0, 1}]

enter image description here

Hope this helps.

  • 2
    $\begingroup$ …and if OP would rather skip the use of an auxiliary variable, there's the option setting Evaluated -> True. $\endgroup$ Aug 11, 2015 at 16:41
  • $\begingroup$ @Guesswhoitis. Is this an option on Plot? I can't seem to find it in 10.2 $\endgroup$
    – Edmund
    Aug 11, 2015 at 17:21
  • $\begingroup$ @Edmund - use Plot[Evaluate[PDF[..., x]], {x, 0, 1}] $\endgroup$
    – Bob Hanlon
    Aug 11, 2015 at 18:49
  • $\begingroup$ Yes, it's an option for Plot[]. $\endgroup$ Aug 12, 2015 at 0:27

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