Joint probability distribution of functions of random variables

[WMDhamnekar]

If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation.

Hint:The joint density function can be obtained by differentiating the following equation with respect to u and v.

$P(U\leq u, V\leq v)=\iint_{(x,y):-(x+y)\leq u,\frac{x}{x+y}\leq v} f_{X,Y} (x,y) dx dy$

Now how to differentiate the above equation with respect to u and v?

 

[WMDhamnekar]

If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation.

Hint:The joint density function can be obtained by differentiating the following equation with respect to u and v.

$P(U\leq u, V\leq v)=\iint_{(x,y):-(x+y)\leq u,\frac{x}{x+y}\leq v} f_{X,Y} (x,y) dx dy$

Now how to differentiate the above equation with respect to u and v?

Hello,

One can get the answer to this question below $\Downarrow$
Joint probability distribution of functions of random variables

 

[HOI]

Leibniz's rule: $$\frac{\partial}{\partial x}\int_{\alpha(x,y)}^{\beta(x,y)} f(x,y,t)dt$$
$$= f(x, y, \beta(x,y))\frac{\partial \beta}{\partial x}- f(x,y, \alpha(x,y))\frac{\partial \alpha}{\partial x}+ \int_{\alpha(x,y)}^{\beta(x,y)} \frac{\partial f(x,y,t)}{\partial x} dt$$.

 

[WMDhamnekar]

Leibniz's rule: $$\frac{\partial}{\partial x}\int_{\alpha(x,y)}^{\beta(x,y)} f(x,y,t)dt$$
$$= f(x, y, \beta(x,y))\frac{\partial \beta}{\partial x}- f(x,y, \alpha(x,y))\frac{\partial \alpha}{\partial x}+ \int_{\alpha(x,y)}^{\beta(x,y)} \frac{\partial f(x,y,t)}{\partial x} dt$$.

Hello,
Would you compute the final answer by using this Leibniz's rule ? In my link, I used differential algebra and PDF of gamma random variable.