I am given this information:
$\tau$ is a constant, $\infty > \tau >0$
Let $X \sim G(n, \lambda)$, where $n \in \{1,2,...,\}$ ($G$ is referring to gamma distribution)
Let $Y = n \tau + X$
Moment generating function of $Y$ (which I derived for the earlier part of the question): $M(t)_{Y} = e^{tn\tau}\left (\frac{\lambda }{\lambda -t} \right )^{n}$
Suppose that $T_{n} = \sum_{i=1}^{n}\left (\tau + Z_{i} \right )$, where $Z_{i} \sim \varepsilon (\lambda)$
For the later part of the question, I need to find the probability density function of $T_{n}$, preferably using the MGF of $Y$
I know that $T_{n} = n\tau+\sum_{i=1}^{n}\left (Z_{i} \right )$ and $Z_{i} \sim G(1,\lambda)$, other than that, I'm lost.
How do you use the MGF of $Y$ to find the PDF of $T_{n}$? Is this the correct approach?