Chapter 6.3, Problem 6E

The joint distribution of amount of pollutant emitted from a smokestack without a cleaning device $\left(Y_1\right)$ and a similar smokestack with a cleaning device $\left(Y_2\right)$ was given in Exercise 5.10 to be

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

The reduction in amount of pollutant due to the cleaning device is given by $U=Y_1-Y_2$

1. a Find the probability density function for U.
2. b Use the answer in part (a) to find E(U). Compare your results with those of Exercise 5.78(c).

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

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

That is. Y₁ and Y₂ are uniformly distributed over the region inside the triangle bounded by $y_1=2,y_2=0$, and $2y_2=y_1$

1. a Find the value of k that makes this function a probability density function.
2. b Find $P\left(Y_1\ge 3Y_2\right)$. That is, find the probability that the cleaning device reduces the amount of pollutant by one-third or more.