Im currently looking for a probabilty density function that posesses the following properties

Should have range (0,1) $$ \lim_{\sigma \rightarrow 0} f(x) = \delta(1) $$ $$ \lim_{\sigma \rightarrow \infty} f(x) = \delta(0) + \delta(1) $$

So far the closest i have gotten is the Beta Distribution which fullfils the second point when $\alpha = \beta$ and they both approach 0.


1 Answer 1


Try a $B(\frac{1}{r},\frac{1}{r+1})$. As $r\to 0$, the first parameter tends to infinity, the second one to $1$, which will approach a point mass at $1$. And as $r\to\infty$, both parameters tend to zero (almost equally quickly), giving you a mixture of two point masses.

beta densities

R code:

xx <- seq(0,1,by=0.0001)
rr <- c(0.01,0.1,1,10,100)
for ( ii in seq_along(rr) ) lines(xx,dbeta(xx,1/rr[ii],1/(rr[ii]+1)),col=ii,lwd=2)
legend("top",lwd=2,col=seq_along(rr),legend=paste("r =",rr))

(I prefer using a parameter $r$ rather than $\sigma$, which connotes a standard deviation.)

  • 1
    $\begingroup$ Thank you this looks really good. I will just leave the question open because I‘m interested in what other ideas people come up with. $\endgroup$
    – elson1608
    Oct 22, 2023 at 20:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.