To find [itex]E |X|[/itex] of a cauchy random variable, I need to integrate(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\int_{-\infty}^{\infty}\frac1{\pi}\frac{|x|}{1+x^2}dx [/itex].

From the definition of absolute value, we have

[itex]\int_{-\infty}^0\frac1{\pi}\frac{-x}{1+x^2}dx + \int_0^{\infty}\frac1{\pi}\frac{x}{1+x^2}dx[/itex] (I think).

But, the very next step in the textbook, which I'm trying to follow along, is

[itex]\frac2{\pi}\int_0^{\infty}\frac{x}{1+x^2}dx[/itex].

Unless I made some mistake, the reasoning here is that

[itex]\int_{-\infty}^0\frac1{\pi}\frac{-x}{1+x^2}dx = \int_0^{\infty}\frac1{\pi}\frac{x}{1+x^2}dx[/itex].

If this is true, why is this?

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# Integrating functions with absolute values

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