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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.

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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)
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.)

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    $\begingroup$ 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

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