Consider the population of 8 million college students. Suppose we are interested in determining \(\mu\), the unknown mean distance (in miles) from the students' schools to their hometowns. We can't possibly determine the distance for each of the 8 million students in order to calculate the population mean \(\mu\) and the population variance \(\sigma^2\). We could, however, take a random sample of, say, 100 college students, determine:
\(X_i\)= the distance (in miles) from the home of student \(i\) for \(i=1, 2, \ldots, 100\)
and use the resulting data to learn about the population of college students. How could we obtain that random sample though? Would it be okay to stand outside a major classroom building on the Penn State campus, such as the Willard Building, and ask random students how far they are from their hometown? Probably not! The average distance for Penn State students probably differs greatly from that of college students attending a school in a major city, such as, say The University of California in Los Angeles (UCLA). We need to use a method that ensures that the sample is representative of all college students in the population, not just a subset of the students. Any method that ensures that our sample is truly random will suffice. The following definition formalizes what makes a sample truly random.
Definition. The random variables \(X_i\) constitute a random sample of size \(n\) if and only if:

the \(X_i\) are independent, and

the \(X_i\) are identically distributed, that is, each \(X_i\) comes from the same distribution \(f(x)\) with mean \(\mu\) and variance \(\sigma^2\).
We say that the \(X_i\) are "i.i.d." (The first i. stands for independent, and the i.d. stands for identically distributed.)
Now, once we've obtained our (truly) random sample, we'll probably want to use the resulting data to calculate the sample mean:
\(\bar{X}=\dfrac{\sum_{i=1}^n X_i}{n}=\dfrac{X_1+X_2+\cdots+X_{100}}{100}\)
and sample variance:
\(S^2=\dfrac{\sum_{i=1}^n (X_i\bar{X})^2}{n1}=\dfrac{(X_1\bar{X})^2+\cdots+(X_{100}\bar{X})^2}{99}\)
In Stat 415, we'll learn that the sample mean \(\bar{X}\) is the "best" estimate of the population mean \(\mu\) and the sample variance \(S^2\) is the "best" estimate of the population variance \(\sigma^2\). (We'll also learn in what sense the estimates are "best.") Now, before we can use the sample mean and sample variance to draw conclusions about the possible values of the unknown population mean \(\mu\) and unknown population variance \(\sigma^2\), we need to know how \(\bar{X}\) and \(S^2\) behave. That is, we need to know:
 the probability distribution of \(\bar{X}\) and \(S^2\)
 the theoretical mean of of \(\bar{X}\) and \(S^2\)
 the theoretical variance of \(\bar{X}\) and \(S^2\)
Now, note that \(\bar{X}\) and \(S^2\) are sums of independent random variables. That's why we are working in a lesson right now called Several Independent Random Variables. In this lesson, we'll learn about the mean and variance of the random variable \(\bar{X}\). Then, in the lesson called Random Functions Associated with Normal Distributions, we'll add the assumption that the \(X_i\) are measurements from a normal distribution with mean \(\mu\) and variance \(\sigma^2\) to see what we can learn about the probability distribution of \(\bar{X}\) and \(S^2\). In the lesson called The Central Limit Theorem, we'll learn that those results still hold even if our measurements aren't from a normal distribution, providing we have a large enough sample. Along the way, we'll pick up a new tool for our toolbox, namely The MomentGenerating Function Technique. And in the final lesson for the Section (and Course!), we'll see another application of the Central Limit Theorem, namely using the normal distribution to approximate discrete distributions, such as the binomial and Poisson distributions. With our motivation presented, and our curiosity now piqued, let's jump right in and get going!