# Joint PDF of sums of exponentially RVs with common terms

Let:

$\gamma_{c_1}=\gamma_1+\gamma_3+\gamma_4$,

and

$\gamma_{c_2}=\gamma_1+\gamma_2+\gamma_4+\gamma_5$,

where $\gamma_i, i=1,...,5$ are i.i.d. exponentially distributed RVs:

$f_{\gamma_i}(\gamma_i)=\frac{1}{\bar\gamma}\exp(-\frac{\gamma_i}{\bar \gamma})$

The sums $\gamma_{c_1}$ and $\gamma_{c_2}$ will hence both be Erlang distributed, with shape parameters $k_{c_1}=3$ and $k_{c_2}=4$.

I need to obtain the joint PDF $f_{\gamma_{c_1},\gamma_{c_2}}(\gamma_{c_1},\gamma_{c_2})$, starting from the conditional

$f_{\gamma_{c_1},\gamma_{c_2}}(\gamma_{c_1},\gamma_{c_2})=f_{\gamma_{c_1}|\gamma_{c_2}}(\gamma_{c_1}|\gamma_{c_2})\times f_{\gamma_{c_2}}(\gamma_{c_2})$.

Since $\gamma_{c_1}$ and $\gamma_{c_2}$ have two common terms ($\gamma_1$ and $\gamma_4$), the joint distribution cannot be written as a product of marginals (i.e., $\gamma_{c_1}$ and $\gamma_{c_2}$ are not independent).

Any ideas how to derive the joint PDF? Thank you.