I need a bit of help with a problem that involves basic knowledge of Poissons Process (distribution).
Given Poissons Process is: $P(x) = \dfrac{\mu^x\cdot e^{-\mu} }{ x!}$ {ie the standard equation} where mew is the mean of the process.
Let $\mu = 5$ (ie. 5 people per month join a tennis club on average)
Also given (but not sure if relevant) that: $P(x<3) = 18.5\cdot e^{-5}$ . (ie. probability that less than 3 people join then tennis club in one month).
Question is: What is the expected waiting time until 3 women join the tennis club, from the start of the period, if the probability that a random person joining is a women is 0.3?
Assume homogeneous poisson process modelles the number of people joining
Thanks!
For a Poisson process, the interarrival time is exponentially distributed with expected value $\frac{1}{\mu}$. However, since only $0.3$ are women, the mean waiting time for a woman to join is $\frac{1}{0.3\mu}$. Let $T_1$ be the time for the first woman to arrive, $T_2$ the section, etc. Then we want the expected value of $T_1+T_2+T_3$ = $\frac{1}{.1\,\mu}=\frac{10}{5}=2$ months