Is there any discrete distribution whose probability mass function resembles beta distribution $f(x; \alpha=5, \beta=1)$

I need a discrete distribution supported on $[0,\infty)$ whose probability mass function is increasing from $P(X = 0) = 0$ to $P(X=x_{0}) = 1$ for a fixed $x_{0}$. (The overall shape can be similar to that of the beta distribution $f(x;1,5)$ Is there such distribution? If the answer is negative, how can I design one with such features?

Solution 1:

Let $p(\alpha, \beta, n; x)$ be the PDF (in $x$) of the beta-binomial distribution with parameters $\alpha$ and $\beta$ on $n$ trials. Then $p(\alpha, \beta, n; nx)$ has support in $[0,1]$ (because the "$nx$" compresses the horizontal axis (number of trials) by a factor of $n$).

Here's the PDF of the $\beta(1,5)$ distribution:

Here's $p(1,5,100;100x)$, the PMF of the corresponding beta-binomial on $100$ trials, with horizontal scaling to $[0,1]$: