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Sampling with replacement is used to find probability with replacement. In other words, you want to find the probability of some event where there's a number of balls, cards or other objects, and you replace the item each time you choose one.

Theorem 6.2. For samples of a single size n, drawn from a population with a given mean ? and variance ?2, the sampling distribution of sample means will have a mean ?¯X=? and variance ?2X=?2n.

Sampling with replacement has two advantages over sampling without replacement as I see it: 1) You don't need to worry about the finite population correction. 2) There is a chance that elements from the population are drawn multiple times - then you can recycle the measurements and save time.

In sampling with replacement the mean of all sample means equals the mean of the population: When sampling with replacement the standard deviation of all sample means equals the standard deviation of the population divided by the square root of the sample size when sampling with replacement.

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