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Some probability density functions' parameters include the mean and the variance (like the normal distribution). Are there examples of probability distributions that are parameterized by other measures of central tendency (median, mode) and dispersion (mean absolute deviation)?

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  • $\begingroup$ One example: If $X$ has the Cauchy distribution $\mathcal C(\mu,\sigma)$ with pdf $\frac{\sigma}{\pi(\sigma^2+(x-\mu)^2)}$ where $x\in\mathbb{R}$, then $\mu$ is indeed the median and $\sigma$ is the Quartile deviation of the distribution. $\endgroup$ – StubbornAtom Feb 17 '18 at 6:23
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I think this is a good question, as it points to an interesting conceptual problem about what a parameter is. In general, if we talk about the probability density function (PDF) of a particular distribution, we usually refer to one established of infinitely many density functions, all of which would describe the distribution equally well. Let me explain this by some examples.

The normal distribution: $\sigma$ versus $\sigma^2$

First, notice that the normal distribution can be parameterized in different ways. At Wikipedia, for instance, you will find the parameterization $$ f_1(x \ | \ \mu,\sigma) \;, \tag{1} $$ where $\mu$ is called the mean and $\sigma$ the standard deviation. But many people prefer $$ f_2(x \ | \ \mu,\sigma^2) \;, \tag{2} $$ where $\sigma^2$ is called the variance. Since $f_1$ and $f_2$ have different signatures, they are two different functions, even though their definitions are identical: $$ f_1(x | \mu, \sigma) \ = \ \dfrac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right) \ = \ f_2(x \ | \ \mu,\sigma^2) \;. \tag{3} $$

The normal distribution: precision $h$

Second, if I remember correctly, Gauß used the parameter $h$, which he called precision, and defined the PDF of the normal distribution as $$ f_3(x \ | \ \mu,h) = \dfrac{h}{\sqrt{\pi}} \exp\left( -h^2(x - \mu)^2 \right) \;. \tag{4} $$ It is yet another parameterization with a different measure of dispersion, and even though (2) looks different from (1), both equations describe the same distribution (to see this, set $h = \dfrac{1}{\sqrt{2\sigma^2}}$).

The normal distribution: mode and median

Third, notice that $\mu$ is not only the mean of the normal distribution, but it is also the mode and the median. So, if you ask for distributions that can be parameterized with the mode or the median, the normal distribution is again an example. It is just a matter of how you interpret $\mu$.

The log-normal distribution: geometric mean $m$

Let me conclude with the log-normal distribution. It is often defined by relating it to the normal distribution:

Iff $\ln(X)$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$, then $X$ follows a log-normal distribution with the same parameters. In symbols: $$ \ln(X) \sim \mathcal{N}\left( \mu, \sigma \right) \,\Leftrightarrow\, X \sim \mathcal{L}\left( \mu, \sigma \right) $$

This is why the following PDF became the established function: $$ g_1(x \ | \ \mu,\sigma) = \dfrac{1}{\sqrt{2\pi\sigma^2x^2}} \exp\left( -\dfrac{(\ln(x) - \mu)^2}{2\sigma^2} \right) \;. \tag{5} $$ Notice, that $\mu$ and $\sigma$ do not correspond to the mean and the standard deviation of the log-normal distribution. Another, and as I think more natural, definition is $$ g_2(x \ | \ m,\sigma^2) = \dfrac{1}{\sqrt{2\pi\sigma^2 x^2}} \exp\left( -\dfrac{\ln^2(\frac{x}{m})}{2\sigma^2} \right) \;, \tag{6} $$ where $m$ corresponds to the geometric mean.

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