Joint distribution of two gamma random variables I am so puzzled by this problem. 
Given two variables $X_1$ and $X_2$, such that $X_i \sim \mathrm{Gam}(a_i, b)$, find the joint distribution of $X_1$ and $X_2$. 
I understand how to proceed if variables are independent, but the general case is a bit unclear. 
Thanks for your help! 
 A: As stated the problem does not make sense, because a joint distribution cannot be found from the marginal distributions! The only meaningful case (as an homework) is to assume independence. In which case the density of the joint distribution is obviously the product of both densities...
A: OP notrockstar knows the solution for the case when the random variables are independent but presumably cannot use it since a solution without the
independence assumption is being sought. Perhaps the OP has posted only 
a simplified version of the question, and what has been left out makes
a solution possible.  For example, if $X_1$ and $X_2$ are given to be the 
times of the $k$-th and $(k+\ell)$-th arrivals in a Poisson process of intensity
(arrival rate) $\lambda$, then these are Gamma random variables with
order parameters $k$ and $k+\ell$ respectively.  Furthermore,
conditioned on $X_1 = x_1$, $X_2$ is a displaced Gamma random variable
with order parameter $\ell$, that is, $X_2 = x_1 + Y$ where
$Y$ is a Gamma random variable with order parameter $\ell$.  Thus,
$$\begin{align*}
f_{X_1,X_2}(x_1,x_2) &= f_{X_2|X_1}(x_2|x_1)f_{X_1}(x_1)\\
&= \begin{cases}
f_Y(x_2-x_1)f_{X_1}(x_1), & 0 < x_1 < x_2 < \infty,\\
0, & \text{otherwise.}
\end{cases}
\end{align*}$$

In view of the additional information provided by the OP that what is really  wanted is the joint distribution of $Y_1 = X_1 + X_2$ and $Y_2 = \frac{X_1}{X_1+X_2}$, maybe the problem is intended as drill in transformation of
variables: can you express the joint density $f_{Y_1,Y_2}(y_1,y_2)$ in
terms of the joint density $f_{X_1,X_2}(\cdot,\cdot)$ as 
$J(y_1,y_2)f_{X_1,X_2}(g_1(y_1,y_), g_2(y_1, y_2))$ with the Gamma
functions thrown in as distractions, or merely as hints that
$X_1, X_2 \in (0, \infty)$ to see if the students can deduce
that $Y_2 \in (0,1)$.
This problem is readily solvable since it is easy to invert
the transformation, find the Jacobian etc.  At the end, one
could say something like "If $X_1$, $X_2$ are assumed to be
independent (this is not stated in the problem given) random
variables with Gamma distributions, then 
the joint density $f_{X_1,X_2}(\cdot,\cdot)$ factors into the product
of the marginal densities, and in this case, 
$f_{Y_1,Y_2}(y_1,y_2)$ equals "$\cdots$" possibly adding that
$Y_1$ and $Y_2$ are obviously independent if they are (I don't
believe they are but am willing to abide a proof that they are),
or giving their marginal pdfs too etc.

In summary, $f_{Y_1,Y_2}(y_1,y_2)$ can be stated in
  terms of the joint density $f_{X_1,X_2}(\cdot,\cdot)$
  without knowing the exact form of $f_{X_1,X_2}$ or the
  marginal densities of $X_1$ and $X_2$.  The assumption that
  $X_1$, $X_2$ are independent can be used at the very end
  to say explicitly what $f_{Y_1,Y_2}(y_1,y_2)$ is; the
  Gammaity or independence of $X_1$ and $X_2$ is not needed
  or used at all in the earlier work, and indeed serves
  merely to clutter up the calculations without shedding
  much light on the matter.

