Take the tour ×  Simple Question:  How can I find a Density
Function of a variable from a Cumulative Function?   Cumulative
Function: $$F(x) = \begin{cases}0 & x < 1 \\ x^2 & 1\leq
x<\infty\end{cases}$$  Density Function = ???  share|improve
this question add comment  1 Answer  up vote 2 down vote
accepted  Note that for some Density Function (DF) $f(x)$, the
Cumulative Density Function (CDF) $F(x)$ is defined as:
Therefore, we can find the DF by differentiating the CDF, i.e.
$f(x)=\frac{d}{dx}F(x)$.  So in your case, where
$$F(x)=\begin{cases}0 & x<1 \\ x^{2} &
x\in[1,\infty)\end{cases}$$ We can find the DF:
$$f(x)=\begin{cases}0 & x < 1 \\ 2x &
x\in[1,\infty)\end{cases}$$  Also just a note: you've tagged
this as probability distributions, but you have not got a valid
probability distribution here as it fails to meet the criterion:
$$\int_{-\infty}^{\infty}f(x)\:dx=\lim_{a\to\infty}F(a)=1$$
share|improve this answer If CDF is 1 - x^(-a), DF is a*x^(-a-1)
? –  Richard Feb 17 at 12:35 Yes, if the CDF is $1-