How to map a histogram by using a suitable distribution?

Question

I want to compute the pdf (Probability density function) for the following distribution, however I am just thinking how to do that? The major problem is: it does not follow any specific distribution.

Answer 1

The only thing possible is an approximation. You'll have to guess at the heights of the bars from the picture. Then you will know the cumulative probability at each division point. You can interpolate linearly between those points.

With that information you can calculate approximate median, mean and standard deviation.

Answer 2

This is the problem of density estimation. There are basically two ways to solve this:

1. Parametric density estimation : assume your density function has a known form with a number of parameters (for instance a normal density where the parameters are the mean and the standard deviation). This can be written as an optimization problem, that can be solved by maximum likelihood estimation for instance.
2. Nonparametric density estimation: don't assume anything abouth the density except your given histogram. For instance, use kernel density estimation.

Of course, in both cases, you get only an approximation of the true density.