Question: Let the pair (X,Y) have a uniform density over the interior of the triangle with vertices at (0,0), (2,0), and (1,2), that is, the density

Let the pair (X,Y) have a uniform density over the interior of the triangle with vertices at (0,0), (2,0), and (1,2), that is, the density is the positive constant in the interior, and zero otherwise.

a. Find the conditional density of Y given X.

b. Find the conditional expectation E[Y|X=1].

c. Find the conditional variance Var[Y|X=1]

d. Assume now that (X,Y) has the above uniform density with probability 1/4, and otherwise has the analogous uniform density over the interior of the triangle with vertices (0,0), (2,0), and (1,4). What is E[Y|X=1] equal to now?

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