Chapter 5.3, Problem 23E

In Example 5.4 and Exercise 5.5, we considered the joint density of Y₁, the proportion of the capacity of the tank that is stocked at the beginning of the week, and Y₂, the proportion of the capacity sold during the week, given by

$f\left(y_1,y_2\right)=\{\begin{array}{l}3y_1,0\le y_2\le y_1\le 1,\\ 0,\text{elsewhere}\text{.}\end{array}$

1. a Find the marginal density function for Y₂.
2. b For what values of y₂ is the conditional density f(y₁|y₂) defined?
3. c What is the probability that more than half a tank is sold given that three-fourths of a tank is stocked?