Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / d / ii / 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 givesAssuming a moment generating function in a neighborhood of zero, write $\mu =\mathbb{E}[Y]$ and $s^2=\mathrm{Var}(Y)$. Conditional expectation and variance yieldThe second contribution to the variance is due to randomness in the number of arriving coaches, not just randomness in the occupancy of each coach.