Combining Random Variables for AP Statistics

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By — McGraw-Hill Professional
Updated on Feb 3, 2011

Practice problems for these concepts can be found at:

Transforming and Combining Random Variables

If X is a random variable, we can transform the data by adding a constant to each value of X, multiplying each value by a constant, or some linear combination of the two. We may do this to make numbers more manageable. For example, if values in our dataset ranged from 8500 to 9000, we could subtract, say, 8500 from each value to get a dataset that ranged from 0 to 500. We would then be interested in the mean and standard deviation of the new dataset as compared to the old dataset.

Some facts from algebra can help us out here. Let μx and σx be the mean and standard deviation of the random variable X. Each of the following statements can be algebraically verified if we add or subtract the same constant, a, to each term in a dataset (X ± a), or multiply each term by the same constant b (bX), or some combination of these (a ± bX):

  • μa ± bX = a ± bμx.
  • σa ± bX = bσX2a ± bx = b2σ2X).
example: Consider a distribution with μX = 14, σX = 2. Multiply each value of X by 4 and then add 3 to each. Then μ3+4X = 3 + 4(14) = 59, σ3+4X = 4(2) = 8.

Rules for the Mean and Standard Deviation of Combined Random Variables

Sometimes we need to combine two random variables. For example, suppose one contractor can finish a particular job, on average, in 40 hours (μx = 40). Another contractor can finish a similar job in 35 hours (μy = 35). If they work on two separate jobs, how many hours, on average, will they bill for completing both jobs? It should be clear that the average of X + Y is just the average of X plus the average for Y. That is,

  • μX ± Y = μX ± μY.

The situation is somewhat less clear when we combine variances. In the contractor example above, suppose that

    σx2 = 5 and σy2 = 4.

Does the variance of the sum equal the sum of the variances? Well, yes and no. Yes, if the random variables X and Y are independent (that is, one of them has no influence on the other, i.e., the correlation between X and Y is zero). No, if the random variables are not independent, but are dependent in some way. Furthermore, it doesn't matter if the random variables are added or subtracted, we are still combining the variances. That is,

  • σ2x ± y = σx2 + σy2 if and only if X and Y are independent.
  • σx ± y = if and only if X and Y are independent.

The rules for means and variances generalize. That is, no matter how many random variables you have: μX1 ± X2 ±... ± Xn = μX1 ± μX2 ± ... + μXn and, if X1, X2,..., Xn are all independent, σ2X1 ± X2 ±…Xn = σ2X1 + σ2X2 +…+σ2Xn.

    example: A prestigious private school offers an admission test on the first Saturday of November and the first Saturday of December each year. In 2002, the mean score for hopeful students taking the test in November (X) was 156 with a standard deviation of 12. For those taking the test in December (Y), the mean score was 165 with a standard deviation of 11. What are the mean and standard deviation of the total score X + Y of all students who took the test in 2002?

solution: We have no reason to think that scores of students who take the test in December are influenced by the scores of those students who took the test in November. Hence, it is reasonable to assume that X and Y are independent. Accordingly,

Practice problems for these concepts can be found at:

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