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

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In summary, "E|X|=E(X)" refers to the expected value of the absolute value of a continuous random variable X. This can be calculated using the formula E(X) = ∫xf(x)dx, where f(x) is the probability density function of X. This equality suggests that the distribution of X is symmetric around its mean, as the absolute value of a number is always positive. It is not applicable to discrete random variables and is significant in probability theory for understanding the central tendency of a continuous random variable.
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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.
 

1. What is the meaning of "E|X|=E(X)?"

The notation "E|X|" refers to the expected value of the absolute value of the random variable X. This is equivalent to the expected value of X when X is a continuous random variable.

2. How is "E|X|=E(X)" calculated?

The formula for calculating the expected value of a continuous random variable X is: E(X) = ∫xf(x)dx, where f(x) is the probability density function of X. To calculate E|X|, we would replace x with |x| in the formula.

3. What does "E|X|=E(X)" tell us about the distribution of X?

"E|X|=E(X)" tells us that the expected value of the absolute value of X is equal to the expected value of X. This implies that the distribution of X is symmetric around its mean, as the absolute value of a number is always positive.

4. Can "E|X|=E(X)" be used for discrete random variables?

No, "E|X|=E(X)" is specifically for continuous random variables. For discrete random variables, the notation is typically written as E[|X|] and the formula for calculating it is different.

5. What is the significance of "E|X|=E(X)" in probability theory?

"E|X|=E(X)" is an important concept in probability theory as it helps us understand the central tendency of a continuous random variable. It tells us the average value of the absolute deviation from the mean, which is useful in analyzing data and making predictions.

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