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By Definition of Expectation of Random Variable:

$$ E(X)= \int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$

Now if the pdf $f_X(x)$ is Even we know that $E(X)=0$ (Ofcourse if integral Converges, i.e, Lets exclude cases like Cauchy Random Variable)

Is the Converse True, i.e., is there a Random Variable $X$ whose pdf is Neither Even-Nor Odd, such that $E(X)=0$.

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    I don't a practical example but probably there is! LOL2012-05-15
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    Have you considered that a probability density function $f_X(x)$ cannot possibly be an **odd** function of $x$? And so your question is whether there exists a random variable whose density function is not an even function of $x$ but whose mean is $0$?2012-05-15

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