Is an expected value of fx(x) the same as this expected value?

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The discussion centers on the expected value calculations for the probability density function fx(x) = 5x^4, defined for 0 < x < 1. The expected value E[x^0.5] is correctly calculated as 10/11 using the integral ∫x^0.5*fx(x) dx from 0 to 1. However, an alternative approach using the transformation Y = X^0.5 leads to a different expected value of 20/33 due to the incorrect application of the transformation. The key insight is recognizing that x is equivalent to y^2, clarifying the relationship between the two variables.

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If a probability density function is fx(x) = 5x^4 for 0<x<1 and 0 elsewhere, then the expected value of E[x^0.5] is the integral of ∫x^0.5*fx(x) between 0 and 1 which is 10/11.

However if I try to approach it as the expected value of Y=X^0.5 then I find fy(y) = 5*lny/y^4 and its expected value is 20/33.

What's wrong? Is it a right approach or are those two things different?
 
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Ah, I got it, x is y^2.. not the other way around.
 

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