Let X be a random variable representing the number of years ofeducation an individual has, and let Y be a random variablerepresenting an individual’s annual income. Suppose that the latestresearch in economics has concluded that:
Y = 6X +U
(1)
is the correct model for the relationship between X and Y , where Uis another random variable that is independent of X. Suppose Var(X)= 2 and Var(Y ) = 172.
a. Find Var(U).
b. Find Cov(X, Y ) and corr(X, Y ).
c. The variance in Y (income) comes from variance in X(education) and U (other factors unobserved to us). What fractionof the variance in income is explained by variance ineducation?
d. How does the fraction you found in (c) compare to corr(Y,X)?
