Lesson 24: Several Independent Random Variables

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pumpkinsIntroduction

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:

  • X1 denote the weight of the first pumpkin sampled
  • X2 denote the weight of the second pumpkin sampled
  • ...
  • Xn denote the weight of the nth 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 X1to affect the weight of another pumpkin, say X2. Therefore, X1X2, ..., Xn 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.

Objectives

  • To get the big picture for the remainder of the course.
  • To learn a formal definition of a random sample.
  • To learn what i.i.d. means.
  • To learn how to find the expectation of a function of n independent random variables.
  • To learn how to find the expectation of a product of functions of n independent random variables.
  • To learn how to find the mean and variance of a linear combination of random variables.
  • To learn that the expected value of the sample mean is μ.
  • To learn that the variance of the sample mean is σ2/n.
  • To understand all of the proofs presented in the lesson.
  • To be able to apply the methods learned in this lesson to new problems.