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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?
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    $\begingroup$ 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. $\endgroup$
    – whuber ♦
    Commented Jan 28, 2020 at 21:06
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    $\begingroup$ 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. $\endgroup$
    – Ben
    Commented Jan 29, 2020 at 6:31

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