# How do I find the probability density function of a random variable X?

Jan 17, 2017

If X~F_X(x), where ${F}_{X} \left(x\right)$ is the probability distribution function, then $F {'}_{X} \left(x\right) = {f}_{X} \left(x\right)$ where ${f}_{X} \left(x\right)$ is the probability density function.

#### Explanation:

By definition

$P \left(X \le x\right) = {F}_{X} \left(x\right)$ where ${F}_{X} \left(x\right)$ is the distribution function of the random variable $X$.

This is sort of analogous to various areas of science where one might consider density as mass divided by volume $\rho = \frac{m}{v}$.

In physics if one were attempting to find how mass is distributed in an object for something like center of mass they would integrate it $x = {\int}_{\Omega} \rho \mathrm{dA} .$

Therein lies the analogy. Just like a physical object is a collection of particles, a probability space is a collection of outcomes.

So, if the probability distribution is described by ${F}_{X} \left(x\right)$, then it would make sense that ${F}_{X} \left(x\right) = {\int}_{\Omega} {f}_{X} \left(x\right) \mathrm{dx}$, where ${f}_{X} \left(x\right)$ is the probability density function.

So,

${F}_{X} \left(x\right) = {\int}_{\Omega} {f}_{X} \left(x\right) \mathrm{dx}$

$\iff$

$F {'}_{X} \left(x\right) = \left({\int}_{\Omega} {f}_{X} \left(x\right) \mathrm{dx}\right) ' = {f}_{X} \left(x\right)$

$\iff$

$F {'}_{X} \left(x\right) = {f}_{X} \left(x\right)$