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I have a model that uses $n$ independent random variables $X_1,..., X_n$. I know the density function of each random variable.

I would like to calculate statistics such as $E(\sqrt{X_1+...+X_n})$ or $Var(X_1 \cdot...\cdot X_n)$.

Is there a computer tool to do it easily rather than finding by myself the correct integrals and use an integral solver?

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    $\begingroup$ I doubt that a general tool is available - outside some fairly special cases, closed-from results are often not available. Depending on the accuracy and explicitness you need, you could sample from the $X_i$ given that you know their density, and then simulate the moments of interest. $\endgroup$ – Christoph Hanck Nov 30 '18 at 9:14
  • $\begingroup$ @ChristophHanck. Thanks. I'm not interested in a closed form, but in a tool that knows how to "transform" these moments to their integral form, and then approximate this integral. $\endgroup$ – Roy Nov 30 '18 at 9:20
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    $\begingroup$ Something like reference.wolfram.com/language/ref/Integrate.html might help. I am still curious how explicit the answers will be in general. Also, for your above examples you would in any case need the joint distribution of the $X_i$ unless they are independent. $\endgroup$ – Christoph Hanck Nov 30 '18 at 10:19
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    $\begingroup$ A solution has been posted on statsSE to the question of the variance of a product of $n$ independent random variables: see stats.stackexchange.com/questions/52646/… ... Difficult to answer otherwise, unless your question is more specific. $\endgroup$ – wolfies Nov 30 '18 at 16:26
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    $\begingroup$ @Wolfies You are the developer of what is arguably the closest thing anyone can find to what is being requested :-). $\endgroup$ – whuber Nov 30 '18 at 21:20