Chapter 5.4, Problem 56E

In Exercise 5.12, we were given the following joint probability density function for the random variables Y₁ and Y₂, which were the proportions of two components in a sample from a mixture of insecticide:

$f\left(y_1,y_2\right)=\{\begin{array}{l}2,0\le y_1\le 1,0\le y_2\le 1,0\le y_1+y_2\le 1,\\ 0,\text{ elsewhere}\text{.}\end{array}$

Are Y₁ and Y₂ independent?

5.12 Let Y₁ and Y₂ denote the proportions of two different types of components in a sample from a mixture of chemicals used as an insecticide. Suppose that Y₁ and Y₂ have the joint density function given by

$f\left(y_1,y_2\right)=\{\begin{array}{l}2,0\le y_1\le 1,0\le y_2\le 1,0\le y_1+y_2\le 1,\\ 0,\text{ elsewhere}\text{.}\end{array}$

(Notice that Y₁ + Y₂ ≤ 1 because the random variables denote proportions within the same sample.) Find

1. a P(Y₁ ≤ 3/4, Y₂ ≤ 3/4).
2. b P(Y₁ ≤ 1/2, Y₂ ≤ 1/2).