Suppose that the inter-arrival times of male customers entering the bank are iid exponential random variables with $\frac{1}{\lambda_1}$ and for those females are iid exponential random variables with $\frac{1}{\lambda_2}$. Find the distribution and expected value of $Z_f$, the number of female customers arriving at the bank between two successive male customers.
The question bothers me for a while. Please help me out!!!
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If $Y$ is the arrival time between successive male customers $Y\sim \text{Exponential}(\lambda_2)$
and interarrival times that are exponential with rate $\lambda_1$ are indicative of a Poisson process with rate $\lambda_1$, therefore the number of female customers arriving given $Y$ is $Z_f|Y\sim \text{Poisson}(\lambda_1Y)$.
I haven't been able to find the distribution as it doesn't seem to marginalize out well, but the expected value is as follows:
$$\begin{split}E(Z_f)&=E(E(Z_f|Y))\\ &=E(\lambda_1Y)\\ &=\frac{\lambda_1}{\lambda_2}\end{split}$$
Conditional the male first arrival time $T$, derive the number of female jumps in $[0, T]$. This the conditional distribution of $Z_f$ on $T$. From here, you can get the marginal of $Z_f$.