EXAMPLE 4.38

Utility bills. Consider a household where the monthly bill for natural-gas averages $125 with a standard deviation of $75, while the monthly bill for electricity averages $174 with a standard deviation of $41. The correlation between the two bills is −0.55.

Let’s compute the mean and standard deviation of the sum of the natural-gas bill and the electricity bill. We let X stand for the natural-gas bill and Y stand for the electricity bill. Then the total is X + Y. Using the rules for means, we have

μ_X+Y = μ_X + μ_Y = 125 + 174 = 299

To find the standard deviation, we first find the variance and then take the square root to determine the standard deviation. From the general addition rule for variances of random variables,

Therefore, the standard deviation is

The total of the natural-gas bill and the electricity bill has mean $299 and standard deviation $63.