# How are probability functions derived? E.g. normal, Poisson, t-distribution

The idea behind PMFs is simple - how likely is a given discrete event to happen? For example, it is intuitive to see why the PMF of a binomial distribution is $$Pr(X=k)= \left( { \begin{array}{} n \\ k \\ \end{array}} \right ) p^k(1-p)^{n-k}$$. This way of writing it arises very naturally when you do $$n$$ independent Bernoulli trials and want to know the probability of $$k$$ successes.

1. But how are the standard PDFs developed?
2. $$\sqrt{2\pi}$$ in the normal distribution formula comes as part of normalizing the integral of the exponential function. But, how can we choose these functions?
3. Can every function with bounded integrals from $$-\inf$$ to $$+\inf$$ be used as a PDF after being scaled appropriately?
• There's a different story behind each distribution. By searching our highest-voted posts (in many cases by including "intuiti*" as a keyword) you will find answers for Normal (Gaussian), Beta, Poisson, Exponential, Uniform, Gamma, Binomial, Bernoulli, and other distributions. In many cases these distributions arise from multiple different considerations: those related to Normal distributions; gap intervals in point processes; various combinatorial processes; distributions related to spheres and balls; maximum-entropy; Central Limit Theorem; stable processes; extreme value processes; and more.
– whuber
Commented Jan 28, 2020 at 21:06
• In answer to your last question, every non-negative function that integrates to a finite value can be scaled to make it a valid density function.
– Ben
Commented Jan 29, 2020 at 6:31