A strange thing my professor told me about expectation

[AxiomOfChoice]

Suppose you know $E[X] = 0$ for a given (continuous) random variable. Does that mean $E[|X|] < \infty$? This is what my professor told me today, though it doesn't really make much sense...

 

[mathman]

It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
On the other hand if the distribution is Cauchy { density 1/[π(1+x²)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.

 

[AxiomOfChoice]

It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
On the other hand if the distribution is Cauchy { density 1/[π(1+x²)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.

Ok, thanks. Looking back in our textbook, the definition of "expectation" begins: "Suppose $\int_\Omega |X(\omega)| dP(\omega) <\infty$. Then we define the expectation..." So I guess that's the way it is!

 

[micromass]

It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
On the other hand if the distribution is Cauchy { density 1/[π(1+x²)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.

If X is Cauchy distributed, then E[X] doesn't exist. So saying E[X]=0 is wrong there. It's as wrong as saying $\int_{-\infty}^{+\infty}{xdx}=0$. The integral simply does not exist.

 

[blue_raver22]

As a scientist, it is important to critically evaluate information and not simply accept it without justification. In this case, it is necessary to understand the concept of expectation and its relationship to a continuous random variable.

The expectation, denoted as E[X], is a measure of the central tendency of a random variable. It represents the average value of the variable over a large number of trials. In the case of a continuous random variable, E[X] is calculated by integrating the variable over its entire range. This means that if E[X] = 0, it indicates that the variable has an equal probability of taking positive and negative values, resulting in a balanced average.

Now, let’s consider the concept of absolute expectation, denoted as E[|X|]. This represents the average absolute value of the random variable, which essentially measures the distance of the variable from its mean. In this case, if E[|X|] is finite, it means that the variable has a finite spread and is not too far from its mean. However, if E[|X|] is infinite, it suggests that the variable has a very large spread and can take extremely high or low values.

Therefore, if E[X] = 0, it does not necessarily mean that E[|X|] is finite. It is possible for a continuous random variable with E[X] = 0 to have a large spread and therefore, an infinite E[|X|]. This is because the absolute value function eliminates the negative values and only considers the magnitude of the variable, which can result in a larger spread.

In conclusion, it is important to understand the underlying concepts and relationships before accepting any statement as true. While E[X] = 0 may suggest a balanced average for a continuous random variable, it does not necessarily imply a finite E[|X|]. Further analysis and understanding of the specific variable and its distribution would be required to make any conclusions about the absolute expectation.