Question

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

Answer 1

If `X~F_X(x)`, where `F_X(x)` is the probability distribution function, then `F'_X(x)=f_X(x)` where `f_X(x)` is the probability density function.

Explanation:

By definition

`P(X<=x)=F_X(x)` where `F_X(x)` 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=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 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(x)`, then it would make sense that `F_X(x)=int_Omegaf_X(x) dx`, where `f_X(x)` is the probability density function.

So,

`F_X(x)=int_Omega f_X(x) dx`

`<=>`

`F'_X(x)=(int_Omega f_X(x) dx)'=f_X(x)`

`<=>`

`F'_X(x)=f_X(x)`