For an applied economics paper, I am looking for a 2-parameter probability distribution function that has the following properties:

  • Simple, closed-form PDF $f(x)$ and CDF $F(x)$, defined on the interval $[0, M]$ (where $M$ may be infinite).

  • $f(x)$ is bell-shaped, in the sense that it is convex and increasing on $[0,a]$, concave and increasing on $[a,f_{max})$, concave and decreasing on $(f_{max}, b]$, and convex and decreasing on $[b, M]$ ($a$, $f_{max}$, and $b$ need not be specified).

  • The expectation of the random variable $X$ following $F(x)$ can be normalized to $1$ by eliminating one of the distribution function's parameters. The other parameter can be used to control the variance.

  • Ideally, it is possible to explicitly solve an equation of the form $A + B F(x) + Cx f(x) = 0$.

So far, the closest distribution I could find is a Gamma distribution with appropriately normalized parameters. However, despite its simplicity, unfortunately no simple closed-form of its CDF exists. The same is true for the Log-Normal and Beta-Distribution.

So my question is: Are there any distributions which I have overlooked so far? Does anything simple and convenient exist that satisfies these criteria?

  • $\begingroup$ I add distribution tag, share the pdf of distribution then we look into it. $\endgroup$
    – SAAN
    Aug 1, 2013 at 13:43
  • $\begingroup$ Curious, why the requirement for a closed-form expression for the CDF? Most software packages have functions to compute both the gamma and lognormal CDFs in an efficient way. $\endgroup$
    – Hong Ooi
    Aug 1, 2013 at 13:54
  • $\begingroup$ One of the equilibrium variables in my paper is defined implicitly by an equation that relates the CDF and PDF of an arbitrary distribution function $F$. I am mainly interested in providing an example where I can solve for the equilibrium variable explicitly. Unfortunately, I can show that other simple distributions like the uniform or exponential do not work, as a solution to my implicit equation can only be found if the probability density is sufficiently high at some (non-zero) point in the distribution. $\endgroup$
    – Martin
    Aug 1, 2013 at 14:07
  • $\begingroup$ I'd split your (revised) first criterion into two. The main reason is that I think that many readers would be interested in distributions satisfying everything else but your own criterion looks very specialised and restrictive at first glance. The principle here is the usual one that this site is not just for short-term attention to new questions but also for long-term building of a repository of threads. $\endgroup$
    – Nick Cox
    Aug 1, 2013 at 15:14
  • $\begingroup$ What do you mean by "solve explicitly"? For instance, would a numerical algorithm work? If not, are you really sure you need some kind of finite algebraic combination of conventional functions? (Usually not: often what you want to do is analyze the properties of the solution and that does not necessarily require a closed form for it.) $\endgroup$
    – whuber
    Aug 1, 2013 at 15:15

2 Answers 2


The log-logistic or Fisk distribution appears to be one candidate.

See http://en.wikipedia.org/wiki/Log-logistic_distribution

It is more flexible than this in so far as it can be J-shaped too.

  • $\begingroup$ Thanks, this looks very promising. I will start to take a closer look at it right now. $\endgroup$
    – Martin
    Aug 1, 2013 at 14:11
  • $\begingroup$ I have checked the distribution now. While it fulfills all of my requirements (at least for certain parameterizations), the CDF and PDF are unfortunately related in such a way that I still cannot obtain a closed-form solution for the equation in my paper (which is of the form $A + B F(x) + C x f(x) = 0$). $\endgroup$
    – Martin
    Aug 1, 2013 at 14:54

Another possibility is the Weibull:


  • $\begingroup$ Thanks, I have looked into it. Similar to the log-logistic distribution, it fulfills all of my requirements, but unfortunately I still cannot get a closed-form solution $x^*$ for the relevant equation in my paper (which is of the form $A+BF(x) + Cxf(x) = 0$). $\endgroup$
    – Martin
    Aug 1, 2013 at 14:58

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