Statistics - Expected Value

Question

A process for producing an industrial chemical yields a product containing two types of impurities.

For a specified sample from this process, let Y1 denote the proportion of impurities in the sample and let Y2 denote the proportion of type I impurities among all impurities found. Suppose that the joint distribution of Y1 and Y2 can be modeled by the following probability density function:

f(y, y2) = 2(1- y2) if 0<=y1<=1 , 0<=y2<=1 and 0 otherwise

Find expected value of the proportion of type I impurities in the sample.

Answer 1

Using Roger's notation, the question asks for the expected value of the sample proportion of chemicals containing type I impurities. So, we should be looking for EX, not EZ = E (X*Y).

(X*Y is the event that impurities of both types are found in the chemical... so, I have no idea why Roger is trying to find E(XY) ???)

Once you know that you are looking for EX, you first find the marginal distribution of X from the joint distribution of X and Y:

f(X) = ∫f(X,Y)dY = ∫₀¹ 2(1-Y)dY = 2Y - Y² |¹₀ = (2 - 1²) - (0 - 0²) = 1

Therefore, EX = ∫₀¹1dX = 1 !!!

In other words, every sample is expected to have the type I impurity!

(This is a bit silly/extreme, which is why I thought the problem had a misprint in the joint distribution of X and Y.)