= Solution
The total is the <independent> random sum $Z_t=\sum_{j=1}^{N_t}Y_j$, with the empty sum zero. Conditional on $N_t=n$, its moment generating function is $M_Y(\theta)^n$. The Poisson probability generating function therefore gives
$$
\boxed{M_{Z_t}(\theta)=\exp\{\lambda t(M_Y(\theta)-1)\}.}
$$
Assuming a moment generating function in a neighborhood of zero, write $\mu=\mathbb E[Y]$ and $s^2=\operatorname{Var}(Y)$. Conditional <expectation> and <variance> yield
$$
\boxed{\mathbb E[Z_t]=\lambda t\mu,\qquad\operatorname{Var}(Z_t)=\lambda t(s^2+\mu^2)=\lambda t\mathbb E[Y^2].}
$$
The second contribution to the <variance> is due to randomness in the number of arriving coaches, not just randomness in the occupancy of each coach.
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