When $X=(X_1,X_2,\ldots,X_K)$ follows a multinomial distribution with the size parameter $N$ and probability vector $p=(p_1,p_2,\ldots,p_K)$, what is the expected number of zero outcomes in $X$?
R code, when $p_i=1/K$ for all $i$, zero outcomes can be simulated as
N <- 100 K <- 50 sum(rmultinom(1,N,rep(1/K,K))==0)
I would like to be able to predict the number of zeros from $N$, $K$, and $p=(p_1,p_2,\ldots,p_K)$. Or I want to know the distribution of the number of zeros.