Part I: Expected value, variance and covariance (8 points in total) This exercis

Question

Part I: Expected value, variance and covariance (8 points in total) This exercise illustrates the properties of expected value, variance and covariance that we are going to use further in this course Note: You solved a similar example during your Section #1. Suppose that for variables X and Y the following holds E(X) = 1 E(Y) =-1 var(X) = 4 var(Y) = 1 cov(X, Y) =-2 Now suppose you generate two new variables equal to (2X+Y) and (X+4Y) (i) Show that E(2X + Y) = 1 and E(X 4Y) =-3 (ii) Show that var(2X + Y) =9 (ii) Show that cov(2X + Y, X + 4Y) =-6 (iv) In part (iii) you showed that cov(2X + Y,X + 4Y)

Explanation / Answer

(iv) This means that the two variables are negatively related, that is, if (2X + Y) increases, then (X + 4Y) will decrease and vice versa. The two variables tend to show opposite behaviour. The correlation between them, if calculated, will also come out to be less than 0.

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