# Distributions with Median=Mode=Average?

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

• Mosly all symmetric distribution, the unimodal symmetric distributions (with expectation existing). – kjetil b halvorsen Feb 26 '15 at 13:19
• A notable example would be any beta distribution where both shape parameters are the same. – Ashe Feb 26 '15 at 13:49
• 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. – Xi'an Feb 26 '15 at 16:24
• 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. – whuber Feb 26 '15 at 19:47
• Although the mean is undefined for it, the Cauchy distribution's median equals its mode. – Alexis Feb 26 '15 at 20:26