Understanding covariance formula

[BHeng9611]

Hi all,

Refer to the highlight in blue box in the attached file, I have 2 questions to ask.

1) May I know why when calculating Cov(X,Y), the first summation is 0.83? Based on the formula given in previous slides, covariance (X,Y) of the population is (1/n)*summation i=1 to n(Xi-μ)(Yi-μ) where n is number of observation. I understand the working (4-6.75)(3-4) part but I dont understand why need multiple 0.3. I dont see a probability term in the formula itself.

2) Regarding the steps that lead up to the answer for Cov (X,Y) = 0.75, I do not see term n being used. Where did term n go?

Seek your kind assistance on these 2 questions.

Thank you in advance,
BX

 

[David Harper CFA FRM]

Hi @BHeng9611

(1) The formula here is (4.0 - 6.75)*(3.0 - 4.00)*30% = 0.8250 which rounds to 0.83. It's contributing to a summation: 0.83 + -0.41 + -0.45 + 0.79 = 0.75 because covariance is the expected cross product. Instead of summing and dividing by (n), we're just multiplying by the probabilities.

(2) Same thing. This is not an ex post (historical) covariance where we have a SAMPLE. This is where we are given the ex ante (population) distributions such that we're computing expected covariance. We don't have any n = sample size, instead we have probabilities! We're relying on the covariance = E(XY) - E(X)*E(Y) per https://en.wikipedia.org/wiki/Covariance Thanks,