Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given i.i.d. draws $x_1,...,x_n$ from $X$, where:

  • $X$ has a finite mean $E[X]=\mu$,
  • $X$ is symmetric about its mean, meaning $f_X(\mu+c)=f_X(\mu-c)$ for all $c$.,
  • The probability density function $f_X$ is not otherwise known.

Is it possible to prove the following?

Proposition. The MLE for the mean of $X$, is the population mean, $\hat \mu_{MLE}=\bar x = \sum_{i=1}^n x_i$.

A proof or a counterexample would be great. I am willing to additionally assume that $X$ has a finite variance $Var[X]=\sigma^2$ if that becomes necessary for the proposition to hold, or if it simplifies the proof.

share|improve this question
Perhaps you can use the invariance property of MLEs in some way? –  hejseb May 13 '14 at 18:26

Your Answer


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

Browse other questions tagged or ask your own question.