Sample as a sequence of random variables

Summary

• Given: a population and a sample $x_1,…,x_n$ . Before implementing my sampling procedure, the sample is unknown . Therefore, $x_i$ (sample point $i$ ) is a random variable . Similarly, $x_1,…,x_n$ is a sequence of random variables.
• With random sampling, $x_1,…,x_n$ will be a sequence of independent and identically distributed (IID) random variables.
• With several measurements from the same sample point, say $(x_i,y_i)$ , we have a random sample if $\left( x_1,y_1 \right),…,(x_n,y_n)$ are independent and identically distributed.
• This implies that any two random variables with distinct index will be independent but it does not imply that $x_i$ and $y_i$ are independent
• It implies that all $x_1,…,x_n$ have the same distribution and that all $y_1,…,y_n$ have the same distribution but it does not imply that $x_i$ and $y_i$ have the same distribution.
• Assuming that our sample is a sequence of random variables from some unknown distribution will make the population irrelevant .