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 .