Relationship of the Poisson-Gamma mixture to the Multinomial distribution

Suppose you have 30 lead acquisition channels, the channels generate leads at the following rates per year,

(lambdas <- round(rgamma(30, shape = 0.8, scale = 20)))

 [1]  5 48 15  8 87 20  2  8  7  3  2  9 22 17  5  2  5 32 27  7  9 55  7  4  2 10 24 16 19 10

In the first year, the first three channels produced the following count of leads,

(X <- Map(function(x) { rpois(1, x) }, lambdas) %>% unlist()) %>% head(3)

[1]  1 51 14

The total number of leads accross all channels in a given year is then Poisson distributed,

(N <- rpois(1, sum(lambdas)))

[1] 504

If,

mu <- lambdas / sum(lambdas)

X could also be said to be multinomial distributed with 504 leads distributed over 30 channels,

ecdf(X) %>% plot()
lines(ecdf(rmultinom(1, size = N, prob = mu) %>% { .[ , 1]}), col = "red")