Question

Answer each of the following questions. You have to show all your work to get full e Problema 1. (5 marks; 3, 2) Assume X ~ Gamma(ai, β) and Y ~ Gamma(a2, β) are independent random variables. a) Compute the joint density of U = X X+Y and VX/X+Y), be sure to include the support/domain.

Answer 1

$Problem~1\\ The~joint~distribution~of~X~and~Y~is\\ f_{X,Y}(x,y)=\frac{\beta^{\alpha_1}x^{\alpha_1-1}\exp(-\beta x)}{\Gamma(\alpha_1)}\times \frac{\beta^{\alpha_2}y^{\alpha_2-1}\exp(-\beta y)}{\Gamma(\alpha_2)}=\frac{\beta^{\alpha_1&plus;\alpha_2}x^{\alpha_1-1}y^{\alpha_2-1}\exp(-\beta (x&plus;y))}{\Gamma(\alpha_1)\Gamma(\alpha_2)}\\ ~since~X~and~Y~are~independent~and~x,y>0\\ =0~otherwise\\ Transform~(X,Y)\rightarrow (U,V)~where\\ u=x&plus;y,~v=\frac{x}{x&plus;y}\Rightarrow x=vu,~y=u(1-v)\\ x,y>0\Rightarrow u>0,~0<v<1\\$

$\begin{vmatrix} J\left(\frac{x,y}{u,v} \right ) \end{vmatrix}=\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{vmatrix}=\begin{vmatrix} v &1-v \\ u& -u \end{vmatrix}=-u,~|Jacobian|=u\\ The~joint~distribution~of~U,~V~is\\ f_{U,V}(u,v)=\frac{\beta^{\alpha_1&plus;\alpha_2}u^{\alpha_1&plus;\alpha_2-1}v^{\alpha_1-1}(1-v)^{\alpha_2-1}\exp(-\beta u)}{\Gamma(\alpha_1)\Gamma(\alpha_2)}\\ =\left(\frac{\beta^{\alpha_1&plus;\alpha_2}}{\Gamma(\alpha_1&plus;\alpha_2)} \exp(-\beta u)u^{\alpha_1&plus;\alpha_2-1}\right )\left(\frac{1}{B(\alpha_1,\alpha_2)}v^{\alpha_1-1}(1-v)^{\alpha_2-1} \right ),\\ u>0,~0<v<1\\ =0~otherwise\\ Hence~U~and`V~are~independently~distributed~where\\ U\sim Gamma(\alpha_1&plus;\alpha_2,\beta)~and~V\sim Beta(\alpha_1,\alpha_2).$