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Suppose the Y = a + bX +U, where X and U are random variables and a and b are constants. Assume that E[U|X] = 0 and that Var[U|X) = X²

a. Is Y a random variable? Why?

b. Is U mean independent of X? Why?

c. Is U independent of X? Why?

d. Show the E[U] = 0 and that Var[U] = E[X²] (10 pts)

e. Show that E[Y|X] = a + bX and that E[Y] = a + bE[X]

f. Show that Var[Y|X] = X² and that Var[Y] = b²Var[X] + E[X²]

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