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It says to use Law of Iterated Expectations, but I don't know how to solve this. Can someone please help me with this question?

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    $\begingroup$ You may get better answers if you show what sort of start you have made even if you think it is wrong then someone can point you in the right direction. $\endgroup$
    – mdewey
    Oct 5, 2017 at 17:25
  • $\begingroup$ You need to follow stats.stackexchange.com/tags/self-study/info. Why the time-series tag? $\endgroup$ Oct 5, 2017 at 19:53

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We have the formula $$ Cov[X,Y] = E[XY] - E[X]E[Y] $$

Compute $$ E[XY] = E[E[XY | X]] = E[X *E[Y|X]] = E[X * (a+bX)] = E[aX + bX^2] $$

$$E[aX+bX^2] = a *E[X] + b * E[X^2]$$

as well as

$$E[Y] = E[E[Y|X]] = E[a+ bX] = a + b *E[X]$$

Subsitute into the formula for covariance, and we find:

$$ Cov[X,Y] = a * E[X] + b * E[X^2] - E[X] * (a+b*E[X])$$ $$ Cov[X,Y] = a * E[X] + b *E[X^2] -a *E[X] -b *E[X]^2$$

$$ Cov[X,Y] = b*E[X^2] -b * E[X]^2 $$ $$ Cov[X,Y] = b * (E[X^2] - E[X]^2) $$ $$ b = \frac{Cov[X,Y]}{Var[X]} $$

since

$$Var[X] = E[X^2] - E[X]^2$$

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  • $\begingroup$ Please don't give full answers to self-study questions $\endgroup$ Oct 6, 2017 at 15:28

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