Hi all. I'm struggling with this HW question. I've searched through the textbook and on the web and have been unable to find a solution(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I've got 4 i.i.d. random variables, X1, X2, X3, X4. Uniformly distributed on [0,1]

so the pdf = 1

and cdf F(x_i) = x_i

Let Y3 = the third highest value of the 4 variables.

What is E(Y3)? I know the answer is 2/5. But I can't figure out how to show this.

2. Relevant equations

I know the formula for E(Y2) (second highest) and E(Y1) (highest), but I cannot find anything about the expectation of the third highest.

[tex] E(Y_1) = \int_{0}^{1} y \cdot 4y^3 \, dy = 4/5 [/tex]

[tex] E(Y_2) = \int_{0}^{1} y \cdot (12y^2 - 12y^3) \, dy = 3/5 [/tex]

based on the pattern, one would assume that E(Y3) = 2/5 (plus I know that's the answer), but I can't derive the formula.

3. The attempt at a solution

I've tried a bunch of different approaches, but they've all been dead ends. Maybe I'm misunderstanding the problem. Or maybe just misunderstanding the concept of order statistics entirely.

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# Expected value of a third order statistic?

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