Question: Suppose that the demands for a company's product in weeks 1, 2, and 3 are each normally distributed. The means are 50, 45, and 65. The standard deviations are 10, 5, and 15. Assume that these three demands are probabilistically independent. This means that if you observe one of them, it doesn't help you to predict the others. Then it turns out that total demand for the three weeks is also normally distributed. Its mean is the sum of the individual means, and its variance is the sum of the individual variances. (Its standard deviation, however, is not the sum of the individual standard deviations; square roots don't work that way.)
a. Suppose that the company currently has 180 units in stock, and it will not be receiving any more shipments from its supplier for at least three weeks. What is the probability that stock will run out during this three-week period?
b. How many units should the company currently have in stock so that it can be 98% certain of not running out during this three-week period? Again, assume that it won't receive any more shipments during this period.