P1.T2 Miller, Chapter 3, Practice question 2

[MissJaguar]

Dear BT Forum,

My understanding is that for computing

COV(X,Y) = E(XY) - E(X)E(Y)

we use joint probability density function to determine E(XY) and individual marginal pdf / pmf / to determine E(X) and E(Y). Why in this practice question the joint probability was used to compute E(X) and E(Y)?

Thanks

 

[David Harper CFA FRM]

Hi @MissJaguar

The source is question T2.304.1 here @ https://forum.bionicturtle.com/threads/p1-t2-304-covariance-miller.6791/

I agree with you (and so does the answer, imo). Please note:

• "Method 2" for the covariance, in fact, employs cov(x,y) = E(xy) - E(x)E(y)
• Re: E(x) and E(y), yes true, but imagine X and Y as a 3*3 matrix, where X is column lablels = [-3%, 1%, 5%] and Y is row labels = [-2, 2%, 3%]: given these joint probabilities (i.e., given only three, that total 100%, instead of nine), only the diagonal has nonzero values [30%,50%,20%] such that these also represent marginal probabilities; e.g., it just happens to be here that: joint prob [x= -3%, y = -2%] = marginal prob [x = -3%] = marginal prob [y = -2%] = 30%. I hope that helps,

 

[MissJaguar]

Dear David,

Very good point. Thanks a million for your answer!