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Calculate $ E(Y) $ when $ Y = max(X, 2\theta - X ) $ when $ X $ ~ $ Uniform U( 0 , 2\theta ) . $

To this question, one of my classmates answered like this Let $ A = X , B = 2\theta - X $

$ E(Max(A,B) = E(\frac{A+B}{2}) +E(\frac{A-B}{2}) $

$ E(Max(A,B) = E(\frac{2\theta }{2}) +E(\frac{2X + \theta }{2}) $

$ E(Max(A,B) = \theta +E(X) + E(\theta) $

$ E(Max(A,B) = 2\theta +E(X) $

And carried on the calculations , now , I dont know this concept of adding and subtracting A,B Can someone please explain the basis of it ???

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  • $\begingroup$ Please add the self-study tag. $\endgroup$
    – mhdadk
    Jul 24, 2021 at 10:04
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    $\begingroup$ see this answer $\endgroup$ Jul 24, 2021 at 10:05
  • $\begingroup$ Hint: $2\theta-Y$ has the same distribution as $Y.$ That leads immediately to the answer with no calculation required. See stats.stackexchange.com/questions/46843 for details. The duplicate addresses your question about "adding and subtracting" random variables. $\endgroup$
    – whuber
    Jul 24, 2021 at 13:56

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