So I recently learned that for a random variable that has Median $=$ Mean, symmetry of the density function around the mean is not implied. I found this to be surprising as the common continuous distributions that I am aware of (Uniform, Normal, etc) that have equal median and mean are symmetric.
I had initially thought of the Gamma distribution as a possibility, however unless I am mistaken, the Mean can approach the Median under certain specifications, but will always be slightly larger.
I am curious if anyone can think of continuous random variable distributions that have an equal median and mean, but are not symmetric about the mean?