In the previous lessons, we explored functions of random variables. We'll do the same in this lesson, too, except here we'll add the requirement that the random variables be independent, and in some cases, identically distributed. Suppose, for example, that we were interested in determining the average weight of the thousands of pumpkins grown on a pumpkin farm. Since we couldn't possibly weigh all of the pumpkins on the farm, we'd want to weigh just a small random sample of pumpkins. If we let:

• \(X_1\) denote the weight of the first pumpkin sampled
• \(X_2\) denote the weight of the second pumpkin sampled
• ...
• \(X_n\) denote the weight of the \(n^{th}\) pumpkin sampled

then we could imagine calculating the average weight of the sampled pumpkins as:

\(\bar{X}=\dfrac{X_1+X_2+\cdots+X_n}{n}\)

Now, because the pumpkins were randomly sampled, we wouldn't expect the weight of one pumpkin, say \(X_1\), to affect the weight of another pumpkin, say \(X_2\). Therefore, \(X_1, X_2, \ldots, X_n\) can be assumed to be independent random variables. And, since \(\bar{X}\) , as defined above, is a function of those independent random variables, it too must be a random variable with a certain probability distribution, a certain mean and a certain variance. Our work in this lesson will all be directed towards the end goal of being able to calculate the mean and variance of the random variable \(\bar{X}\). We'll learn a number things along the way, of course, including a formal definition of a random sample, the expectation of a product of independent variables, and the mean and variance of a linear combination of independent random variables.