Question

A gun shop sells gunpowder. Monthly demand for gunpowder is normally distributed, average 20 pounds, and has a standard deviation of 2 pounds. The shop manager wishes to stock gunpowder inventory at the beginning of each month so that there is only a 2% chance that the shop will run out of gunpowder. (i.e., that demand will exceed inventory) in any given month. Calculate the amount of gunpowder to stock in inventory, in pounds.

Answer 1

Let X be the random variable denoting the amount of gunpowder in the shop.

X ~ N(20, 2) i.e. (X - 20)/2 ~ N(0,1)

Let a be the amount (in pounds) of gunpowder to stock in inventory.

Thus, P(X < a) = 1 - 0.02 = 0.98 i.e. P[(X - 20)/2 < (a - 20)/2] = 0.98 i.e. [(a - 20)/2] = 0.98 i.e. (a - 20)/2 = (0.98) = 2.054 i.e. a = 20 + (2 * 2.054) = 24.108 pounds.

So, the shop manager needs to stock 24.108 pounds of gunpowder.