What is the PDF of a product of a geometric random variable and an exponential random variable?

Let $X$ be a geometric random number, and $Y$ be an exponential random number. Then PDF of $X$ will be $$f_X(x)=p(1-p)^x$$ and $$f_Y(y)=\frac{1}{t}\exp\left(-\frac{y}{t}\right).$$ Let $Z=XY$, then what is the PDF of a new random number? Thank you.

The best way to do this via MGF. Let $\phi$ be the MGF of $X$
$Ee^{uXY}=E[E[e^{uXY}|X]]=E[E[e^{uxY}]|_{x=X}]=E[(\frac{1/t}{1/t-u})^Y]=\phi(\log(\frac{1/t}{1/t-u}))=\frac{p/t}{p/t-u}$
that means the distribution of $XY$ is ? with parameter $p/t$