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Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.

Edit 2

We know that $\mu$ is a population characteristic. Let us assume the population is of size $n$. Denote by $Y$ the random variable that describes the population characteristic. Then $\mu=\text{E}(Y)$. Let $Y_1,\ldots,Y_n$ denote the random variables of respectively individual 1 to $n$. By definition $\mu=\frac{\sum_{i=1}^nY_i}{n}$. We then make a sample from the population. How can we obtain a sample of size $n+1$ or $n\rightarrow\infty$ that is iid from a population that is finite? Here some say that we can sample from $Y_1,\ldots,Y_n$ WITH replacement.

Edit 3

Of course, if we consider a sample of size $N$ with $N\leq n$, where the sampling is done without replacement, then we can obtain a iid sample if the original RVs are independent. WHY? If $Y_1,\ldots\ Y_n$ are iid and we let $X_1=Y_1,\ldots, X_n=Y_n$ and consider any subset of $X_1,\ldots, X_n$, then this subset will consist of iid RVs.

Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.

Edit 2

We know that $\mu$ is a population characteristic. Let us assume the population is of size $n$. Denote by $Y$ the random variable that describes the population characteristic. Then $\mu=\text{E}(Y)$. Let $Y_1,\ldots,Y_n$ denote the random variables of respectively individual 1 to $n$. By definition $\mu=\frac{\sum_{i=1}^nY_i}{n}$. We then make a sample from the population. How can we obtain a sample of size $n+1$ or $n\rightarrow\infty$ that is iid from a population that is finite? Here some say that we can sample from $Y_1,\ldots,Y_n$ WITH replacement.

Edit 3

If we consider a sample of size $N$ with $N\leq n$, where the sampling is done without replacement, can we can obtain a iid sample if the original RVs are independent? If $Y_1,\ldots\ Y_n$ are iid and we let $X_1=Y_1,\ldots, X_n=Y_n$ and consider any subset of $X_1,\ldots, X_n$, then this subset will consist of iid RVs, right? Am I missing some point here?

Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.

Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.

Edit 2

We know that $\mu$ is a population characteristic. Let us assume the population is of size $n$. Denote by $Y$ the random variable that describes the population characteristic. Then $\mu=\text{E}(Y)$. Let $Y_1,\ldots,Y_n$ denote the random variables of respectively individual 1 to $n$. By definition $\mu=\frac{\sum_{i=1}^nY_i}{n}$. We then make a sample from the population. How can we obtain a sample of size $n+1$ or $n\rightarrow\infty$ that is iid from a population that is finite? Here some say that we can sample from $Y_1,\ldots,Y_n$ WITH replacement.

Edit 3

Of course, if we consider a sample of size $N$ with $N\leq n$, where the sampling is done without replacement, then we can obtain a iid sample if the original RVs are independent. WHY? If $Y_1,\ldots\ Y_n$ are iid and we let $X_1=Y_1,\ldots, X_n=Y_n$ and consider any subset of $X_1,\ldots, X_n$, then this subset will consist of iid RVs.

Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean

$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$.

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.

Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.