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Suppose $X$ and $Y$ are normally distributed with known means and standard deviations. How do I calculate what is the probability distribution some function $f(X,Y)$? For example, say $f(X,Y)=2X+Y$. How does one do this kind of computations?

user82298
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  • Hint: What is the distribution of the sum of two Gaussian random variables? Is it not a Gaussian, as well ? – kaka Jul 14 '15 at 18:07
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    As @Mark noted, there is not sufficient information to define the $f(X,Y)$. You need to know something about the relation between $X$ and $Y$. A common assumption would be independence, which would allow full characterization $f(X,Y)$. If this assumption is allowed, I would suggest editing the post to reflect this. – Cliff AB Jul 14 '15 at 18:12
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    @kaka No. In general the sum of 2 Gaussian random variables is not Gaussian as well. If X and Y are Bivariate Normal, then the sum of X and Y is Gaussian, but in general not. This is a common fallacy. – Mark L. Stone Jul 14 '15 at 18:15
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    What if $Y = - 2 X$? When you want to answer a very general question that may not really have one it's usually a good strategy to try to come up with simple counterexamples. – dsaxton Jul 14 '15 at 19:25

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There is insufficient information to make calculations f(X,Y). The dependency, if any, between X and Y determines their joint distribution, and hence any function of X and Y. See my answer and Glen_b's comment to my answer at Distribution for an operation of variables with identical distributions .

If it is known that X and Y are jointly Normal, i.e., Bivariate Normal, then calculations could be made if you also know the correlation coefficient between X and Y, or equivalently since means and standard deviations are known, cov(X,Y). If X and Y are Bivariate Normal, you can use the formulas in https://en.wikipedia.org/wiki/Multivariate_normal_distribution for the case n = 2. In particular, if X and Y are Bivariate Normal, then X + 2*Y is Normal, and E(2*X + Y) = 2*E(X) + E(Y), and Var(2*X + Y) = 4*Var(X) + Var(Y) + 4*Cov(X,Y).

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