0
$\begingroup$

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?

$\endgroup$
2
  • 1
    $\begingroup$ Why not (horizontally) scale the Beta-binomial distribution with the same parameters? (Admittedly, the tailing off is a little slower...) $\endgroup$ Nov 25 at 7:05
  • $\begingroup$ @EricTowers: Can you expand a bit about the scaling idea you noted? $\endgroup$
    – User
    Nov 25 at 7:57
1
$\begingroup$

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: Mathematica graphics

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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