Is there any important distribution with Median=Mode=Mean, apart from the Gaussian and Student's distribution?

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    $\begingroup$ Mosly all symmetric distribution, the unimodal symmetric distributions (with expectation existing). $\endgroup$ Commented Feb 26, 2015 at 13:19
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    $\begingroup$ A notable example would be any beta distribution where both shape parameters are the same. $\endgroup$
    – Ashe
    Commented Feb 26, 2015 at 13:49
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    $\begingroup$ It certainly works for all scale mixtures of Gaussian distributions, since$$f(x)=\int_0^\infty\tau\varphi(\tau \{x-\mu\})\text{d}\pi(\tau).$$This obviously includes the Student's $t$ distributions. $\endgroup$
    – Xi'an
    Commented Feb 26, 2015 at 16:24
  • $\begingroup$ What does "important" mean in this question? That word is rather subjective and vague. Incidentally, distributions with this property do not need to be symmetric. $\endgroup$
    – whuber
    Commented Feb 26, 2015 at 19:47
  • $\begingroup$ Although the mean is undefined for it, the Cauchy distribution's median equals its mode. $\endgroup$
    – Alexis
    Commented Feb 26, 2015 at 20:26

1 Answer 1


The Laplace and the Logistic I believe are the other two most well-known, from the category "univariate continuous unimodal" which appears to be the focus of interest.

The Beta distribution with equal shape parameters has been mentioned in a comment -the qualification is that the parameters must be higher than unity. If they are exactly unity we obtain the Uniform distribution (which is not unimodal), and if they are lower than unity, we obtain an interesting "mirror" case with many applications that has acquired a name of is own, namely the Arcsine distribution, in the sense that here mean=median = minimizer, rather than the mode (maximizer).

  • $\begingroup$ OK, I accept this answer because is the only one posted as "answer" but all your contributions are great. Thanks. $\endgroup$
    – skan
    Commented Feb 27, 2015 at 0:50

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