# Distribution of number of times units are chosen for a simple random sample

Simple random sampling with replacement.

Population size = $N$

Sample size = $n$

I am interested in learning about the distribution of the random variables $X_0, X_1, ..., X_n$, where $X_i$ is the number of units in the population that were chosen exactly a number $i$ of times for the sample. Note that: $$\sum_{i=0}^n X_i = N$$

If three and only three units in the population have been choosen just four times for the sample, then $X_4 = 3$ . $X_0,X_1,...,X_n$ are absolute frequencies of absolute frequencies.

$X_0$ is the number of units in the population that are not chosen for the sample. If $X_n = 1$, then $X_0 = N−1$ and $X_1,X_2,...,X_{n−1}=0$

Is it possible to find out the functional form of the joint probability mass function of $X_0,X_1,...,X_n$?

• If you have $X_0$, $X_1$, .... $X_n$, your sample size is not $n$ Commented Aug 29, 2017 at 3:57
• No. That is not correct. A unit in the population can be chosen for the sample 0, 1, 2, .... or n times. If three and only three units in the population have been choosen just four times for the sample, then $X_4 = 3$. $X_0,X_1,...,X_n$ are absolute frequencies of absolute frequencies. Commented Aug 30, 2017 at 7:55
• $X_0$ is the number of units in the population that are not chosen for the sample. If $X_n = 1$, then $X_0 = N - 1$ and $X_1, X_2, ..., X_{n-1} = 0$. Commented Aug 30, 2017 at 8:27
• Sorry; I see. Your question could be a little clearer though. Commented Aug 30, 2017 at 11:27
• [OT] is it a problem from statistical mechanics? looks like some occupancy distribution ... Commented Aug 30, 2017 at 12:39

Suppose $x_r \ge 0$, $\sum_{r=0}^n x_r = N$ and $\sum_r r x_r = n$.
The joint probability for observing $X_r = x_r$ all $r$ is: the number of allocations of the population into appropriately sized buckets * the number of label allocations * the probability of each $n$-sized sample realisation, $${N \choose x_0 \ldots x_n} \frac{n!}{\prod_{r=0}^n (r!)^{x_r} } (1/N)^n$$