Question 3.EX.47: If E[Y |X] = 1, show that Var(X Y) ≥ Var(X)...
If E[Y |X] = 1, show that
Var(X Y) ≥ Var(X)
E[X^{2}Y^{2}|X] = X^{2}E[Y^{2} |X]\\ \geq X^{2}(E[Y|X])^{2} = X^{2}
The inequality follows since for any random variable U, E[U^{2}] ≥(E[U])^{2} and this remains true when conditioning on some other random variable X. Taking expectations of the preceding shows that
E[(XY)^{2}] \geq E[X^{2}]As
E[XY] = E[E[XY |X]] = E[XE[Y |X]] = E[X]
the results follow.
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