# Calculating statistics on transformations of many random variables [closed]

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?

## closed as off-topic by kjetil b halvorsen, Michael Chernick, mkt, gung♦Jan 29 at 15:25

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• 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. – Christoph Hanck Nov 30 '18 at 9:14
• @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. – Roy Nov 30 '18 at 9:20
• 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. – Christoph Hanck Nov 30 '18 at 10:19
• 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. – wolfies Nov 30 '18 at 16:26
• @Wolfies You are the developer of what is arguably the closest thing anyone can find to what is being requested :-). – whuber Nov 30 '18 at 21:20