# 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?

• Why not (horizontally) scale the Beta-binomial distribution with the same parameters? (Admittedly, the tailing off is a little slower...) Nov 25 at 7:05
• @EricTowers: Can you expand a bit about the scaling idea you noted?
– User
Nov 25 at 7:57

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 $$[0,1]$$: 