Hi, I am stuck on this question, especially part i). Please try to answer atleas

Question

Hi, I am stuck on this question, especially part i). Please try to answer atleast part i)

Y is a random variable that equals one when an individual goes to college and zero when he or she does not. X is another random variable that captures the individual’s income group. Specifically, X = 1 if the individual is in the lowest income group, and the values 2, 3 and 4 indicate the next 3 groups of respectively higher incomes. The joint distribution of X and Y is as follows.

(g) Do individuals who go to college have a higher mean of the income
group variable than individuals who do not attend college? Is the
income group distribution more "spread" out for individuals who
attend college or for ones that don’t? Explain

(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 the independence.

(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?

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

Explanation / Answer

Two events are said to be independent if P(X=x, Y=y) = P (X = x) * P(Y=y)

Since we need a strictly positive probability distribution, none of the outcomes can have 0 probability.

This will be true for the following distribution:

Thus, P ( X = 1, Y = 1 ) = 1/8

P (X = 1) = 1/4

P (Y = 1) = 1/2

Thus, P (X = 1) * P (Y = 1) = 1/4 * 1/2 = 1/8

This shows that the events are independent.

Hope this helps.

X = 1 X = 2 X = 3 X = 4 Total Y = 0 1/8 1/8 1/8 1/8 1/2 Y = 1 1/8 1/8 1/8 1/8 1/2 Total 1/4 1/4 1/4 1/4 1

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