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Let $X$ and $Y$ be any two random variables.(a) Show that $E[\operator name{Var}(Y | X)]=E\left[Y^{2}\right]-E \mu_{Y Y}^{2}$ . IHint: Use the variance shortcut formula and apply the Law of Total Expectation to the first term.(b) Show that $\operatorname{Var}(E[Y | X])=E \mu_{Y|X}^{2}-(E[Y])^{2} .[H i n t :$ Use the variance shortcut formula again; this time, apply the Law of Total Expectation to the second term.](c) Combine the previous two results to establish the Law of Variance.

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

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in this exercise were given to random variables X and Y, and for part a were asked to show the following. We can do this by beginning with the variant shortcut formula, the conditional variance for why given X So this is why given X squared. Okay, so, so far I've just written out the variant shortcut formula for the conditional variance of why given X and now using the law of total Expectation on the first term, this is equal to the expectation of why squared and the second term is equal to the expectation of the mean of why given X and that's squared. So that's part a next for Part B were asked to show the following. Now, using the variants shortcut formula, we can rewrite this as follows. So the second line is just applying the variant shortcut formula for the variance on the conditional expectation of why Given X now, the first term can be rewritten like this. And for the second term, if we apply the law of total expectation, we get the following. That's just the expectation on why, and then it's all squared. So that completes Part B and then for part C. We're asked to combine the results from part A and part B to show the law of total variance. So if we take the left side of the equations for both part A and part B and then add them, we will get the following. So using our results we have and this comes out of the following and using this is basically just the variant shortcut formula. So this is equal to the variance, the conditional variance of why given X and so here we have shown the law of total variance by equating these two things.

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