Relationship between f(x) and F(x)

A continuous random variable's probability density function, \(f(x)\), and its cumulative distribution function, \(F(x)\) are related by:

\[ F(x) = \int_0^x f(u) \;du \spaced{and} f(x)= F'(x) \]

This can sometimes be used to find the distribution of a random variable that is defined as a function of one or more others.

  1. Find the cumulative distribution function of \(X\), \(F(x) = P(X \le x)\).
  2. Differentiate it to get the probability density function, \(f(x)= F'(x)\)

Question: Square root of an exponential variable

Consider a random variable, \(X\), with an exponential distribution

\[ X \;\;\sim\;\; \ExponDistn(\lambda) \]

What is the distribution of \(Y = \sqrt{X}\)?

(Solved in full version)