example of a uniform density function

Question

Could you, please, give an example (in biology, physics, economics etc.) of a constant density function, i.e. I would like to know the example of a density function $f_{X}(x)$ such that $f_{X}(x) = const$ for $x\in (a,b)$ for some $a < b \in \mathbb{R}$.

Answer 1

I suppose the ultimate example of a 'flat' density functions is the uniform family. For example a uniform distribution on the interval $(0,1)$ often denoted $\mathsf{Unif}(0,1)$ or $\mathsf{Beta}(1,1).$ You can look at the uniform and beta families of distributions on Wikipedia.

Sometimes Bayesian statisticians want to use prior distributions that contain very little information. Then one workable idea is to use a distribution that has a very large variance. Perhaps something like $\mathsf{Norm}(\mu = 0, \sigma=1000),$ which spreads most of its probability very 'thinly' over the interval $(-3000, 3000);$ it's variance is $\sigma^2 = 1000^2.$ This distribution does not have a constant value over any interval, but is "almost" flat everywhere.

If you can say why you want a flat distribution, I (or someone else here) might be able to give you an answer specifically attuned to your purpose.