Expected area of rectangle

Question

This question is taken from Rice - Mathematical statistics.

A random rectangle is formed in the following way: The base, X, is chosen to be a uniform [0, 1] random variable and after having generated the base, the height is chosen to be uniform on [0, X]. Use the law of total expectation, Theorem A of Section 4.4.1, to find the expected circumference and area of the rectangle.

I have trouble understanding the answer for finding the expected area of rectangle.

Let H be the height of the rectangle. The area is XH, so

$E(XH) = E(E(XH|X)) = E(XE(H|X)) = 1/2 E(X^2)$

Can someone kindly explain how do you derive the 3rd part from the 2nd part of the equation ?

It's a basic property of conditional expectation. Haven't you encountered that in the book yet? — Cave Johnson, Nov 01 '16 at 08:59
You may check this wikipedia page. "If $X$ is $\mathcal{H}$-measurable, then $E(XY\mid\mathcal{H})=XE(Y\mid\mathcal{H})$." — Cave Johnson, Nov 01 '16 at 09:03

score 1 · Accepted Answer · answered Nov 01 '16 at 09:59

1

Do you know what $E(XH|X)$ means? It is expected value of $XH$ if we know $X$. So X is a constant, and $E(cH)=cE(H)$ - this is a common property of expectation.

answered Nov 01 '16 at 09:59

Djura Marinkov

2,595

Expected area of rectangle

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