Properties of Random Samples

For the next two weeks, I will discuss some of the concepts of random sample which we use very frequently. These are certainly not the central focus of this course, but it is extremely important for all of us to know these concepts. We have to use these ideas throughout this quarter. First we need to know what do we mean by a random sample. Definition: The random variables X1, ..., Xn together is known as the random sample of size n from the population f(x|θ) if X1, ..., Xn are mutually independent, or the joint density of X1, ..., Xn is given by ∏n i=1 f(xi|θ). We will commonly write as X1, ..., Xn iid ∼ f . Example: X1, ..., Xn is a random sample from exponential(β). What is P (X1 ≤ a1, ..., Xn ≤ an). Note that