probability-distribution that has its mode equal median

Question

Could anyone tell me any asymmetric distribution whose mode=median? Thanks in advance.

Answer 1

Lots of distributions do. Any continuous distribution which is symmetric about its largest pdf value will satisfy this.

Student T and Laplacian are two simple examples

Answer 2

One can produce examples, though not necessarily interesting ones. Let $X$ have density function $f(x)=0$ for $x\lt 0$, $f(x)=x$ for $0\le x\le 1$, and $f(x)=e^{-2(x-1)}$ for $x\gt 1$. The median and the mode are at $x=1$. The distribution is very much not symmetric about $x=1$.

We can also produce simple discrete examples. Let $\Pr(X=0)=\frac{1}{2}$. Let $\Pr(X=1)=\Pr(X=2)=\frac{1}{8}$ and $\Pr(X=-47)=\Pr(X=-99)=\frac{1}{8}$.

If you feel like it, you can modify the discrete example and get an asymmetric distribution where mean, median, and mode are all equal.