Exercise Solution 4.1

The distinction between a random sample and a realization of a random sample parallels the distinction between a random variable and a specific realization of that random variable.

Given a random vector (or random variable) X, a random sample is a set {X[1]X[2], … , X[m]} of random vectors (or random variables) that are independent and all have the same distribution as X. Intuitively, we think of the sample as a set of random vectors (random variables) representing m independent draws from the distribution of X. A realization of this sample is a set {x[1]x[2], … , x[m]} of vectors (or numbers) all in the range of X. Intuitively, we think of them as being one result of, m times, randomly drawing from the distribution of X. Specifically, if we think of X[k] as representing the kth random draw from the distribution of X, then we think of x[k] as being the result of that random draw.

Suppose random variable X represents the result of tossing a single 6-sided die. A random sample {X[1]X[2]X[3],X[4]X[5]} would then represent the result of tossing that die five times. A realization of that random sample might be {4, 3, 6, 6, 1}.

A random sample is a set of random vectors (random variables). A realization of a sample is a set of vectors (numbers).