Question

2.42 Scooping ice cream. Ice cream usually comes in 1.5 quart boxes (48 fluid ounces), and ice cream scoops hold about 2 ounces. However, there is some variability in the amount of ice cream in a box as well as the amount of ice cream scooped out. We represent the amount of ice cream in the box as X and the amount scooped out as Y. Suppose these random variables have the following means, standard deviations, and variances: nean SD variance X 48 20.25 0.0625 (a) An entire box of ice cream, plus 3 scoops from a second box is served at a party. How much ice cream do you expect to have been served at this party? What is the standard deviation of the amount of ice cream served? (b) How much ice cream would you expect to be left in the box after scooping out one scoop of ice cream? That is, find the expected value of X -Y. What is the standard deviation of the amount left in the box? (c) Using the context of this exercise, explain why we add variances when we subtract one random variable from another.

Answer 1

C)

When we subtract one random variable from other is an example of something random happening, and when that happens, the uncertainity increases whether we add to the pile or subtract from it. Here we have to find the variance, i.e. the variability in the amount of ice cream left in the box when we scoop out one scoop of ice cream. The amount of variability i.e. variance would increase and cannot decrease when we scoop out one scoop of ice cream as the uncertainity has increased due to the combination of the two random variables.

Also if individual random variable variances were to subtracted, overall variance could become negative which is not possible.