# Finding the probability density function of one random variable in multiple random variable

The question looks like this, where fx is equal to this:

$f(x) = exp(-(1/2(1 - \rho^2 - \gamma^2))(1-\gamma^2)x_1^2 + x_2^2 + (1-\rho^2)x_3^2 -2\rho x_1 x_2 +2 \rho \gamma x_1 x_3 -2 \gamma x_2 x_3))/(2\pi)^3/2(\sqrt(1-\rho^2 - \gamma^2)$

I need to find the probability density function of $X_1$ alone and $X_2 + X_3$ together. I know how to find probability density function of $X_1$ alone which is integrating the above expression wrt to $X_2, X_3$. But not sure how to find probability density function of $X_2 + X_3$.

Moreover, I need to find joint density function of $X_1$ and $X_2 + X_3$. I am not sure how to do this as well.

Is there any way of finding the density function without integrating wrt to $X_1$ or $X_2 + X_3$?

Need some guidance on solving this.

• can you place a better picture or type it in. – Caran-d'Ache Sep 27 '13 at 9:21
• hope it better... – lakesh Sep 27 '13 at 9:36

First of all you do not need to differentiate the above expression but rather integrate it with respect to the variables which you need to get rid of. Lets call $\vec{X}=\{X_1,X_3,X_3\}$ You have $f_{\vec{X}}(x_1,x_2,x_3)$. Then using the standard rules of transformations of random variables one can get: $$f_{X_1}(x_1)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{\vec{X}}(x_1,x_2,x_3)\; \mathrm dx_3 \mathrm dx_2$$ Then $$f_{X_2,X_3}(x_2,x_3)=\int_{-\infty}^{\infty}f_{\vec{X}}(x_1,x_2,x_3)\; \mathrm dx_1$$ and $$f_{Y=X_2+X_3}(y)=\int_{-\infty}^{\infty}f_{X_2,X_3}(x_2,y-x_2)\mathrm dx_2$$ And $$f_{X_1,\;Y=X_2+X_3}(x_1,y)=\int_{-\infty}^{\infty}f_{\vec{X}}(x_1,x_2,y-x_2)\mathrm dx_2$$
• just a qn: the last equation you need to integrate wrt to both $dx_2$ and $dx_1$ right? – lakesh Sep 27 '13 at 15:20
• shouldn't the third equation be an integral of $f_x2(x_2)f_x3(y-x_2)dx_2$? – lakesh Sep 27 '13 at 16:22
• @lakesh both times "no". In the last eqn you seek for joint pdf, so you get rid of only $x_2$ (or $x_3$). In the third eqn you can not factor pdf's (because $x_2$ and $x_3$ are not independent), so again you have to use joint pdf. – Caran-d'Ache Sep 27 '13 at 17:21
• but u need to integrate wrt $x_1$ right? – lakesh Sep 27 '13 at 17:24