Homework SourseSuppose that Y a bX U where X and U are random variables an
Suppose that 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 that E[U]=0 and that Var[U]=E[X2]
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]
Solution
![Suppose that 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? Wh](/WebImages/3/suppose-that-y-a-bx-u-where-x-and-u-are-random-variables-an-976123-1761500725-0.webp "Suppose that 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? Wh")
