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.
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)