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Suppose that X and Y are continuous and jointly distributed by f(x, y) = c(x +...

Suppose that X and Y are continuous and jointly distributed by f(x, y) = c(x + y)2 on the triangular region defined by 0 ≤ y ≤ x ≤ 1.
a. Find c so that we have a joint pdf.
b. Find the marginal for X
c. Find the marginal for Y.
d. Find E[X] and V[X].
e. Find E[Y] and V[Y].
f. Find E[XY]
g. Find cov(X, Y).
h. Find the correlation coefficient for the two variables.
i. Prove that P(X + Y ≤ 1) = 3/14 by setting up the definite integral (s) which lead you to it.
j. Find E[Y|X].
k. The least square regression line for Y over X is defined to be β1 ∗ X + β0 where
i. β1 = cov(X,Y)/V[X]

ii. β0 = E[Y] − β1 ∙ E[X]
Do you think this is a good prediction of E[Y|X] based on a value of X?

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