# Drawing $m$ objects from $n$ types (with replacement). What is the distribution of type counts?

I am drawing $m$ objects randomly from $n$ types (with replacement). Then I count how many times each type is selected. What is the expected distribution of counts of each type?

More precisely, let $c_k$ be the count of objects of type $k$ (where $k=1\dots n$). What is the probability distribution of $c_k$?

My second question is about the distribution of counts. More precisely, let $q_c$ be the number of types that get counted $c$ times:

$$q_c = \sum_{k=1}^n \delta_{c_k,c}$$

where $\delta_{x,y}$ is Kronecker's delta.

What is the expected value of $q_c$?

• – Clement C. Mar 22 '16 at 14:48
• @ClementC. Thank you, I solved it using this. – becko Mar 23 '16 at 17:24

I am drawing $m$ objects of $n$ types (with replacement). Let $c_{k}$ be the count of objects of type $k$ ($k=1,\dots,n$). We have: $$\sum_{k=1}^{n}c_{k}=m$$ and the probabilities of counts are given by:
$$P\left(c_{1},\dots,c_{n}\right)=\frac{m!}{n^{m}}\frac{1}{c_{1}!\dots c_{n}!}$$ Applying the multinomial theorem, we find that the marginal probability that $c_{k}=c$ is:
$$p_{k}\left(c\right)=\binom{m}{c}\frac{\left(n-1\right)^{m-c}}{n^{m}}$$
and it follows that $\left\langle c_{k}\right\rangle =m/n$. Let $q_{c}$ be the number of types that get selected $c$ times: $$q_{c}=\sum_{k=1}^{n}\delta_{c_{k},c}$$ By noting that $\left\langle \delta_{c_{k},c}\right\rangle =p_{k}\left(c\right)$, we find: $$\left\langle q_{c}\right\rangle =\binom{m}{c}\frac{\left(n-1\right)^{m-c}}{n^{m-1}}$$