202405301312
Status: #idea
Tags: Probability Theory
State: #nascient

Probability Density Functions

We say $f$ is the p.d.f for a random variable $X$ if it is possible to write it as:

Observe this forces:

1. $f\ge 0$
2. $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx=1$, since $P(-\mathrm{\infty },\mathrm{\infty })=1$

While we are used to that being always the case in elementary probability it is not always possible.

References