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 1/2, 2 tickets with probability 2/5 and 3 tickets with probability 1/10.

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

Attempted Solution:
Number of customers in the first hour: N_f(1)+N_m(1) = N_c(1) ~ Poi(20+30).
Number of customers that buy i ticket: N_i = N_c(1)\times P(Buy i tickets)