Bounded Distribution with specific limits regarding Variance

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.

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.

R code:

xx <- seq(0,1,by=0.0001)
rr <- c(0.01,0.1,1,10,100)
plot(c(0,1),c(0,10),type="n",las=1,xlab="",ylab="")
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.)

• 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. Oct 22, 2023 at 20:20