Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $E(X)=0$.

share|cite|improve this question
I don't a practical example but probably there is! LOL – checkmath May 15 '12 at 12:10
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 $0$? – Dilip Sarwate May 15 '12 at 12:13

It is possible to make up as many examples as you wish. Let $Y$ be almost any random variable with mean $\mu$, and let $X=Y-\mu$. All we need to do is to avoid symmetry about $\mu$, so for example let $Y$ have exponential distribution, or density $2y$ in $[0,1]$ and $0$ elsewhere.

For a discrete example, let $Y$ have binomial distribution with $p\ne 0$, $1$, or $1/2$.

share|cite|improve this answer
very good answer Andre, thanks :) – Ekaveera Kumar Sharma May 15 '12 at 12:59

Define $f_X(x):=\frac 1{x^3}\chi_{[1,+\infty)}(x)+\frac 12\chi_{(-5/2,-3/2)}$.

  • Since $\int_{\Bbb R}f_X(x)dx=\left[-\frac 1{2x^2}\right]_1^{+\infty}+\frac 12=1$ and $f_X$ is non-negative, it's a density.
  • We have \begin{align} E[X]&=\int_1^{+\infty}\frac 1{x^2}dx+\frac 12\int_{-5/2}^{-3/2}xdx\\\ &=\left[-\frac 1x\right]_1^{+\infty}+\frac 12\left[\frac{x^2}2\right]_{-5/2}^{-3/2}\\ &=1+\frac 14\frac 14(9-25)\\ &=0. \end{align}
  • $f_X$ is not even, since $f_X(10)=\frac 1{1000}\neq 0=f_X(-10)$

(there should be simpler examples)

Note that a density function cannot be odd, since in this case $\int_{-\infty}^{+\infty}f(x)dx=0$, whereas it should be $1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.