# Expected coverage after sampling with replacement 'k' times [duplicate]

If I sample with replacement $k$ times from a jar of with a finite population of $N$ unique marbles. What is the probability distribution for the fraction of the unique marbles that I sample?

-

## marked as duplicate by Byron Schmuland, tomasz, Chris Eagle, Norbert, Noah SnyderOct 7 '12 at 19:09

For $i=1$ to $N$, let random variable $X_i$ be $1$ if $i$ is chosen at least once, and let $X_i=0$ otherwise.
The probability that $X_i=1$ is $1$ minus the probability that the number is chosen no times. On any one trial, the probability of not choosing $i$ is $\frac{N-1}{N}$. Hence $$\Pr(X_i=1)=1-\left(\frac{N-1}{N}\right)^k.$$ The number $Y$ of $i$ chosen is given by $$Y=\sum_{i=1}^N X_i,$$ so by the linearity of expectation, $$E(Y)=\sum_{i=1}^N X_i=N\left( 1-\left(\frac{N-1}{N}\right)^k \right).$$ For the expected proportion, divide by $N$.