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Let $X$, $Y$, $Z$ be independent random variables that have a (positive) joint density $f_{XYZ}$ with respect to the Lebesgue measure on $\mathbb{R}^3$, with positive marginals $f_X$, $f_Y$, $f_Z$.

How do we obtain the density $(X,Y,Z)$ conditional on $X+Y+Z$?

The joint density of $(X,Y)$ conditional on $X+Y+Z$ seems to be doable based on the strategy in this question: Conditional density of Sum of two independent and continuous random variables but in this question, the change of variables and the explicit calculation of the Jacobian seems to be important, and I don't know how to apply this here.

Example using e.g. Gaussian random variable would be much appreciated.

kisten
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1 Answers1

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Take $E_k=\{x+y+z=k\}\cap \mathbb{R}^3$. The conditional pdf of $(X,Y,Z)$ given that $(X,Y,Z)\in E_k$ is $$f_{XYZ|E_k}(x,y,z)=\frac{f_{XYZ}(x,y,z)\cdot 1_{E_k}(x,y,z)}{\iint_{E_k}f_{XYZ}\mathrm{d}S}$$ In fact, if $A\subseteq\mathbb{R}^3$ is arbitrary, then $$\mathbb{P}\left((X,Y,Z)\in A|(X,Y,Z)\in E_k \right)=\frac{\iint_{E_k\cap A}f_{XYZ}\mathrm{d}S}{\iint_{E_k}f_{XYZ}\mathrm{d}S}$$

Matthew H.
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  • Thank you for your answer. Are you saying that $\mathbb{P}(E_k)=\iint_{E_k}f_{XYZ}\mathrm{d}S$? But $X+Y+Z$ is (e.g. for $X$, $Y$, $Z$ being independent Gaussian) just a Gaussian random variable. Why don't you have $\mathbb{P}(E_k)=\mathbb{P}({X+Y+Z=k})=0$? – kisten Aug 09 '22 at 12:35
  • This does not say $\mathbb{P}(X+Y+Z=k)$ equals $\iint_{E_k}f_{XYZ}\mathrm{d}S$. You should try to show using this formula that if $X,Y,Z\sim \exp(\lambda)$ are iid, and if $k>0$, then the $$(X,Y,Z)|E_k\sim \mathcal{U}(E_k)$$ – Matthew H. Aug 09 '22 at 13:32
  • My question is more of how to derive this formula; cf. https://math.stackexchange.com/questions/473790/conditional-density-of-sum-of-two-independent-and-continuous-random-variables. I get that you should have the surface measure on $E_k$ as $E_k$ is a lower dimensional surface in $\mathbb{R}^3$ and thus Lebesgue measure zero there, but I don't see how you derive that. – kisten Aug 09 '22 at 14:24