# Partial convolution of random variables in random vectors

Suppose $(X_1,Y_1)$ is a bivariate random vector with a given distribution and let $X_2$ be a univariate random variable with the same distribution as $X_1$ but independent of $(X_1,Y_1)$. I want to find the distribution of $(X_1+X_2,Y_1)$.

As an specific example suppose $(X_1,Y_1)$ is bivariate normal so my first attempt was the following code:

\[ScriptCapitalD] =
TransformedDistribution[{u + w ,
v}, {{u, v} \[Distributed]
BinormalDistribution[{0, 0}, {1, 1}, 0.5],
w \[Distributed] NormalDistribution[0, 1]}];

PDF[\[ScriptCapitalD], {x, y}]

PDF[
TransformedDistribution[{u + w,
v}, {{u, v} \[Distributed]
BinormalDistribution[{0, 0}, {1, 1}, 0.5],
w \[Distributed] NormalDistribution[0, 1]}], {x, y}]

Mean[\[ScriptCapitalD]]

{0, 0}

Covariance[\[ScriptCapitalD]]

{{2., 0.5}, {0.5, 1.}}


I have tried some other distributions such as uniform distribution as well but I can not get the PDF of the joint distribution. Although I can find the answer myself by using the transformation theory for random variables I wonder if it is possible to get the results in Mathematica.

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Nothing to add to resolution, other than it's interesting that things like RandomVariate work fine on the transformed distribution. Puzzling, perhaps a bug. –  rasher Mar 31 at 0:05
I'm sure you've noted the same result using an equivalent MultinormalDistribution in the transformed distribution: PDF returns unevaluated, even for numeric values, yet other functions on it seem to be fine. Perhaps time for a ping to WRI support. –  rasher Mar 31 at 1:07