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

1 Answer
256
Jan 17, 2017

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)#

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