I would have thought the answer to the following was another Binomial distribution, but I can't seem to get Mathematica to output that fact:
PDF[TransformedDistribution[x1 + x2, {x1, x2} \[Distributed] BinomialDistribution[n, p]], y]
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2 Answers
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Your syntax is slighlty off. The way you wrote it, {x1, x2} \[Distributed] BinomialDistribution[n, p]] indicates that the vector variable {x1, x2} follows the multivariate distribution BinomialDistribution[n, p], which of course does not work.
Instead, you need to indicate the distribution for each variable:
PDF[TransformedDistribution[
x1 + x2, {x1 \[Distributed] BinomialDistribution[n, p],
x2 \[Distributed] BinomialDistribution[n, p]}], y]
This is shown in the second syntax example in the documentation for TransformedDistribution.
Bob Hanlon also pointed out that a more readable result can be obtained by evaluating the TransformedDistribution itself:
TransformedDistribution[x1 + x2,
{x1 \[Distributed] BinomialDistribution[n, p],
x2 \[Distributed] BinomialDistribution[n, p]}
]
(* Out: BinomialDistribution[2 n, p] *)
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2$\begingroup$ Evaluation of just
TransformedDistribution[ x1 + x2, {x1 \[Distributed] BinomialDistribution[n, p], x2 \[Distributed] BinomialDistribution[n, p]}]givesBinomialDistribution[2 n, p]which makes it clear that the result is aBinomialDistributionrather than having to visually recognize the fact from thePDF. And on my system, your input evaluates toPiecewise[{{(1 - p)^(2*n - y)*p^y* Binomial[2*n, y], 0 <= y <= 2*n}}, 0]$\endgroup$ Commented Jul 16, 2018 at 15:17 -
$\begingroup$ @Bob Both are excellent points, thank you. I added the result of evaluation of
TransformedDistributionas it is indeed more readable, and fixed the output image, which I had copied wrong. $\endgroup$– MarcoBCommented Jul 16, 2018 at 16:09 -
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{x1, x2} \[Distributed] ...would be fine with multinomial distributions. The form ofDistributedmust follow each distribution individually, though. $\endgroup$– kirmaCommented Jul 16, 2018 at 16:12 -
$\begingroup$ @kirma I don't understand your point. Could you elaborate on your second sentence? $\endgroup$– MarcoBCommented Jul 16, 2018 at 17:01
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$\begingroup$ For instance, look at
MultinomialDistributiondocumentation. Single value distributions work with single variable, others demand list of correct length... but you can't trivially combine them. $\endgroup$– kirmaCommented Jul 16, 2018 at 17:47
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For joint distribution of n iid random variables each with distribution d, you can also use ProductDistribution[{d, n}].
TransformedDistribution[x1 + x2,
{x1, x2} \[Distributed] ProductDistribution@{BinomialDistribution[n, p], 2}]
BinomialDistribution[2 n, p]
PDF[dist, y] // TeXForm
$\begin{cases} p^y \binom{2 n}{y} (1-p)^{2 n-y} & 0\leq y\leq 2 n \\ 0 & \text{True} \end{cases}$
