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Suppose that ($X$, $Y$ and $Z$) are three random variables with joint pdf \begin{align} f(x,y,z)=\begin{cases} \frac{1}{\pi}\exp(x(y+z-x-2))-\frac{1}{2}(y^2+x^2), & x\geq 0,\; x\in R, \; y\in R \\ 0, & \text{oterwise} \end{cases} \end{align} Ι have to find the joint conditional distribution of $Y$ and $Z$, given $X=x$.

My problem is that I can't solve the integral above. \begin{align} F(z,y|X=x)=\int_{0}^{y} \int_{0}^{z} \frac{1}{\pi}\exp(x(k+t-x-2))-\frac{1}{2}(k^2+t^2)\,\mathrm{d}k\,\mathrm{d}t \end{align}

Can anybody help me?

Björn Friedrich
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Spy93
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    If $x$ is fixed and $g(y,z):=f(x,y,z)$ then $g(y,z)$ is "almost" a PDF of the conditional distribution that you mention. It only fails when it comes to the condition $\int\int g(y,z)dydz=1$, but this can be repaired by dividing $g(y,z)$ by the constant $\int\int g(y,z)dydz$. It is not necessary to find the conditional CDF as you yield. – drhab Jul 13 '17 at 13:22

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$$f_{Y,Z}(y,z|X=x) = \frac{f_{X,Y,Z}(x,y,z)}{f_{X}(x)}$$

Find the marginal $f_{X}(x) = \int_{y}\int_{z}f_{X,Y,X}(x,y,z)dzdy$ and divide joint pdf by it.

Dhruv Kohli
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