Chapter 5.3, Problem 28E

In Exercise 5.10, we proved that

$f\left(y_1,y_2\right)=\{\begin{array}{l}1,\text{ 0}\le y_1\le 2,0\le y_2\le 1,\text{ 2}{y}_2\le y_1,\\ 0,\text{ elsewhere}\end{array}$

is a valid joint probability density function for Y₁, the amount of pollutant per sample collected above the stack without the cleaning device, and for Y₂, the amount collected above the stack with the cleaner.

1. a If we consider the slack with the cleaner installed, find the probability that the amount of pollutant in a given sample will exceed .5.
2. b Given that the amount of pollutant in a sample taken above the stack with the cleaner is observed to be 0.5, find the probability that the amount of pollutant exceeds 1.5 above the other stack (without the cleaner).

5.10 An environmental engineer measures the amount (by weight) of particulate pollution in air samples of a certain volume collected over two smokestacks at a coal-operated power plant. One of the stacks is equipped with a cleaning device. Let Y₁ denote the amount of pollutant per sample collected above the stack that has no cleaning device and let Y₂ denote the amount of pollutant per sample collected above the stack that is equipped with the cleaning device.

Suppose that the relative frequency behavior of Y₁ and Y₂ can be modeled by

$f\left(y_1,y_2\right)=\{\begin{array}{l}k,\text{ 0}\le y_1\le 2,0\le y_2\le 1,\text{ 2}{y}_2\le y_1,\\ 0,elsewhere.\end{array}$

That is, Y₁ and Y₂ are uniformly distributed over the region inside the triangle bounded by y₁ = 2, y₂ = 0, and 2y₂ = y₁

1. a Find the value of k that makes this function a probability density function.
2. b Find P(Y₁ ≥ 3Y₂). That is, find the probability that the cleaning device reduces the amount of pollutant by one-third or more.