有放回与无放回抽样。

https://web.ma.utexas.edu/users/parker/sampling/repl.htm

1、有放回抽样,对于样本之间是独立的。数学上表示,有放回抽样的样本与原样本集合,协方差为0. 即

When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second. Mathematically, this means that the covariance between the two is zero.

无放回抽样,

In sampling without replacement, the two sample values aren't independent. Practically, this means that what we got on the for the first one affects what we can get for the second one. Mathematically, this means that the covariance between the two isn't zero.

In particular, if we have a SRS (simple random sample) without replacement, from a population with variance 统计与抽样-保持愤怒, then the covariance of two of the different sample values is 统计与抽样-保持愤怒, where N is the population size.

 

2、当样本总量足够大时,这两种抽样差别不大。

When we sample without replacement, and get a non-zero covariance, the covariance depends on the population size. If the population is very large, this covariance is very close to zero. In that case, sampling with replacement isn't much different from sampling without replacement.

 

3、无放回抽样会导致 抽样出的样本 方差变小。这也是为什么方差为什么除以N-1。

“分母是n-1才能使对方差的估计(而不是方差)是无偏的”

https://blog.csdn.net/weixin_41776824/article/details/80548039

“为什么方差的分母是n-1?
结论: 这个问题本身概念混淆了。如果已知全部的数据,那么均值和方差可以直接求出。但是对一个随机变量X,需要估计它的均值和方差,此时才用分母为n-1的公式来估计他的方差,因此分母是n-1才能使对方差的估计(而不是方差)是无偏的。因此,这个问题应该改为,为什么随机变量的方差的估计的分母是n-1?”