$X$~$Bin(n,p),Y_n$~$N(μ,\sigma^2)$
Where X is the number of trials taking place, and $Y_n$ is the amount of time the $n$th trial takes (independent of other trials).
$Z$ is a new random variable that denotes the total time of all trials taking place.
I believe $Z$ to be $\sum\limits_{n=1}^X Y_n$.
The problem is I have no idea where to start when finding the probability for situations with $Z$. eg. $P(Z<60)$
Do I find the joint p.d.f. first? $f(x,y)=f(x)f(y)$?
Any hints are appreciated! Thanks!
Try characteristic function: we know the CF for $N(\mu,\sigma^2)$ is $$\phi(t)=\exp\left\{i\mu t-\dfrac{1}{2}\sigma^2t^2\right\}$$ \begin{align} \mathbb{E}[e^{itZ}]&=\mathbb{P} [X=0]+\mathbb{E}\left[\exp \left\{it\sum\nolimits_{j=1}^kY_j\right\}\cdot\mathbb{1}_{X\geq 1}\right]\\ &=\mathbb{P} [X=0]+\sum\nolimits_{k=1}^n\mathbb{E}\left[\mathbb{1}_{X=k}\cdot\prod\nolimits_{j=1}^k \exp\{it Y_j\}\right]\\ &=\mathbb{P}[X=0]+\sum\nolimits_{k=1}^n\mathbb{P}[X=k]\phi^k(t) \hspace{10pt}\text{(by independence of $X$, $Y_1$, $Y_2$, $\cdots$)}\\ &=\sum_{k=0}^n\mathbb{P}[X=k]\phi^k(t)\ =\ \sum_{k=0}^n\binom{n}{k}p^k\phi^k(t)(1-p)^{n-k}\\ \end{align} Then we can recover the density of $Z$ from CF. The computation is a little complicated, but this is the only method I know. Maybe some other users have better idea!