Question

Exercise 1. An automatic beverage vending machine has a random quantity ?2 of drink in stock at the beginning of a given day and dispenses a random quantity ?1 for the day (with amounts expressed in gallons). The machine is not resupplied during the day and, in

consequence ?1 ≤ ?2. It has been observed that ?1 and ?2 have a joint density given by

f(y1,y2)={1/2 0<=y1<=y2<=2,}

{0 in other point }

This, the points (?1, ?2) are evenly distributed in the triangle with the given borders.
Find the conditional density of ?1 given ?2 = ?2. Evaluate the probability that it will be sold less than 1⁄2 gallon, since the machine contains 1.5 gallons at the beginning of the day; and also if contains 2 gallons at the start of the day.

Answer 1