# 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

## 1 Answer

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 $$f_{X_1,\;Y=X_2+X_3}(x_1,y)=\int_{-\infty}^{\infty}f_{\vec{X}}(x_1,x_2,y-x_2)\mathrm dx_2$$

• Sorry made a mistake when i typed. – lakesh Sep 27 '13 at 9:39
• 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