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Simple Question:

How can I find a Density Function of a variable from a Cumulative Function?

Example:

Cumulative Function: $$F(x) = \begin{cases}0 & x < 1 \\ x^2 & 1\leq x<\infty\end{cases}$$

Density Function = ???

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1 Answer 1

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:

$$F(x_{1}):=\int_{-\infty}^{x_{1}}f(x)\:dx$$

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: $$\int_{-\infty}^{\infty}f(x)\:dx=\lim_{a\to\infty}F(a)=1$$

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If CDF is 1 - x^(-a), DF is a*x^(-a-1) ? –  Richard Feb 17 '13 at 12:35
    
Yes, if the CDF is $1-x^{-a}$ then the DF is $\frac{d}{dx}(1-x^{-a})=ax^{-(a+1)}$ –  Shaktal Feb 17 '13 at 12:42

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