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Y is an random variable that equals one when an individual goes to college and zero when he or she does not. X is another random vari- able that captures the individual's quartile in the income distribution. Specically, X = 1 if the individual is in the rst (lowest) income quartile, and the the values 2, 3 and 4 indicate the next 3 quartiles, respectively. The joint distribution of X and Y is as follows.

X = 1 X = 2 X = 3 X = 4
Y = 0 6/30 8/30 4/30 2/30
Y = 1 1/30 2/30 4/30 3/30

(a) What is the marginal probability of X? What is the mean of X? What is the variance of X?

(b) What is the marginal probability of Y ? What is the mean of Y ? What is the variance of Y ?

(c) What is the conditional probability of Y = 0 given X = 2?

(d) What is the distribution of Y conditional on X = 2, i.e. what is the probability of each possible outcome of Y conditional on X = 1? What is the distribution of Y conditional on X = 2? ...and on X = 3 and on X = 4? What is the distribution of X conditional on Y = 0?

(e) What is the mean value of X conditional on Y = 0? This value is called the conditional mean of X (given Y = 0).

(f) What is the variance of X conditional on Y = 0? This value is called the conditional variance of X (given Y = 0).

(g) Does the joint distribution indicate that the mean income quar- tile is higher for individuals who attend college or ones that don't? Does it indicate that the income quartile distribution more spread out" for individuals who attend college or ones that don't? Ex- plain.

(h) What is covariance of X and Y ? What is the correlation of X and Y ? Are X and Y independent?

(i) Provide a new joint distribution of X and Y such that they have a strictly positive probability of every pair of outcomes, and such that X and Y are independent. Demonstrate that, given the new joint distribution function, X and Y are independent.

(j) Which joint distribution coincides more closely with your intuition: the original one below or the new one you provided? Why? Do you suspect that X causes Y , or that Y causes X, or both?

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