Problem 38E

Let Y1 denote the weight (in tons) of a bulk item stocked by a supplier at the beginning of a week and suppose that Y1 has a uniform distribution over the interval 0 ≤ y1 ≤ 1. Let Y2 denote the amount (by weight) of this item sold by the supplier during the week and suppose that Y2 has a uniform distribution over the interval 0 ≤ y2 ≤ y1, where y1 is a specific value of Y1.

a Find the joint density function for Y1 and Y2.

b If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter-ton?

c If it is known that the supplier sold a quarter-ton of the item, what is the probability that she had stocked more than a half-ton?

Solution

Step 1 of 3

a) Let Y1 is the weight of the bulk

And Y1 has a uniform distribution with the interval means (0,1)

Then

Let Y2 is the amount of the item sold

And Y2 has a uniform distribution with the intervalmeans(0,y1)

Then

We know

=

=

=

Hence