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(y1, y2) = { k , 0≤y1≤2, 0≤y2≤1, 2y2≤y1 {0, elsewhere.
That is, Y1 and Y2 are uniformly distributed over the region inside the triangle bounded by y1 = 2, y2 = 0, and 2y2 = y1.
a Find the value of k that makes this function a probability density function.
b Find P(Y1 ≥ 3Y2). That is, find the probability that the cleaning device reduces the amount of pollutant by one-third or more.