E(X) and the Equality of Expectation: Debunked or Confirmed?

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SUMMARY

The discussion centers on the equality of expectation, specifically whether E|X| equals E(X). The example provided uses the random variable X = {-1, 0, 1}, calculating E(X) as 0 and E|X| as 2/3. The conclusion drawn is that E|X| does not equal E(X), confirming that the equality does not hold in this case. This analysis clarifies a common misconception in probability theory regarding the expectation of absolute values.

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torquerotates
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Is this true? In all cases?
 
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E(X) = expectation of X

Let X = {-1, 0, 1}
E(X) = (-1+0+1)/3 = 0

E|X| = ( |-1|+|0|+|1| )/3 = (1+0+1)/3 = 2/3

So, I think no, it's not true that E|X| = E(X) if I understand the question correctly.
 

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