Question: f(y₁,y₂)= 1, 0 ≤ y₁ ≤ 2, 0 ≤ y₂ ≤ 1, 2y₂ ≤ y₁,

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

a. Find E(Y₁) and E(Y₂).

b. Find V(Y₁) and V(Y₂).

c. The random variable Y₁-Y₂ represents the amount by which the weight of pollutant can be reduced by using the cleaning device. Find E(Y₁ – Y₂).

d. Find V(Y₁ –Y₂). Within what limits would you expect Y₁ –Y₂ to fall?