Females and males arrive at times of independent Poisson processes with rates 30 and 20 per hour.

Regardless of sex, customers buy 1 ticket with probability $\displaystyle 1/2,$ $\displaystyle 2$ tickets with probability $\displaystyle 2/5$ and $\displaystyle 3$ tickets with probability $\displaystyle 1/10.$

Let $\displaystyle N_i$ be the number of customers that buy $\displaystyle i$ tickets in the first hour. Find the joint distribution of $\displaystyle (N1, N_2, N3).$

Attempted Solution:

Number of customers in the first hour: $\displaystyle N_f(1)+N_m(1) = N_c(1)$ ~ $\displaystyle Poi(20+30).$

Number of customers that buy i ticket: $\displaystyle N_i = N_c(1)\times P(Buy$ $\displaystyle i$ $\displaystyle tickets)$