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Suppose that the number of women who buy concert tickets follows a Poisson process with rate $30$ women per hour, and similarly the number of men who buy tickets is a Poisson distribution with rate $20$ per hour.
Regardless of gender customers buy

  • $1$ ticket with probability $1/2$
  • $2$ tickets with probability $2/5$
  • $3$ tickets with probabiltiy $1/10$

Letting $N_i$ be the number of customers that buy $i$ tickets in the first hour, find the joint distribution of $(N_1,N_2,N_3)$.

I know that since the number of females $N_F$ and $N_M$ are independent, that their superposition $N_F+N_M$ is a Poisson process with rate $30+20=50$, and also that the arrival times follow an ordered uniform distribution. But I don't what a joint distribution on the vector $(N_1,N_2,N_3)$ would even look like. Any help is appreciated.

Attempt.

Since $N_1+N_2+N_3=N(1)$ is a Poisson process with rate $50$, then $$N_1\sim Poisson(\frac{1}{2}\cdot 50)$$ is a Poisson process, and similarly for $N_2$ and $N_3$, $$N_2\sim Poisson(\frac{2}{5}\cdot 50)$$ $$N_3\sim Poisson(\frac{1}{10}\cdot 50)$$ So the joint distribution function is given by $$P(N_1=i_1,N_2=i_2,N_3=i_3)=\prod_{k=1}^3e^{-50p_k}\dfrac{(50p_k)^{i_k}}{i_k!}$$ where $p_k$ is the probability that someone buys $k$ tickets, for $k=1,2,3$.

JB071098
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    This should be called as splitting Poisson process. At the end you should be able to show that the are independent Poisson. See, e.g. https://math.stackexchange.com/questions/1777427/splitting-poisson-process-formal-proof – BGM Mar 28 '19 at 12:14
  • Could you check to see if I have the right idea? @BGM – JB071098 Mar 28 '19 at 12:27
  • Yes that should be the desired joint pmf. Now remaining is whether you need to supplement intermediate calculations steps as some sort of proof, or you can just directly quote the result like this. – BGM Mar 29 '19 at 03:36

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