# 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

*. Intuitively, we think of the sample as a set of random vectors (random variables) representing*

**X***m*independent draws from the distribution of

*. A realization of this sample is a set {*

**X**

*x*^{[1]},

*x*^{[2]}, … ,

*x*^{[m]}} of vectors (or numbers) all in the range of

**. Intuitively, we think of them as being one result of,**

*X**m*times, randomly drawing from the distribution of

*. Specifically, if we think of*

**X**

**X**^{[k]}as representing the

*k*

^{th}random draw from the distribution of

*, then we think of*

**X**

*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).