Let X be random variable with cumulative distribution function
$P(X\leq x) = \left\{ \begin{array}{ll} 0, \quad \quad x<0,\\ \frac{x}{8}, \quad \quad 0\le x < 2 \\ \frac{x^2}{16},\quad \quad 2\le x< 4\\ 1, \quad \quad x \ge4
\end{array} \right.$
I know $f(x)= \frac{d}{dx} F(X) $
How do we find the PDF in this case..
Another question in my mind is that , Since CDF is just area under the curve, So we can find many function which give the same area for that interval. Will the pdf obtained from the CDF be unique?
If I am wrong do correct me ..
Thank you…
Best Answer
You have stated that the CDF is the area under the curve...which curve? The answer is the PDF.
In other words, if the CDF $F(x)$ is the area under the curve for some function $f(x)$ it means that $$ F(x)=\int_{-\infty}^x f(t) dt $$ for some function $f(t)$, so that $F'(x)=f(x)$ by the FTC. The function $f(x)$ is the PDF.
In practice, you only have to differentiate the CDF to obtain the PDF.
NOTE: This operation is unique. In the other direction, integrating the PDF to obtain the CDF would be unique up to a constant of integration, which would be obtained by the condition $$ \int_{-\infty}^\infty f(x) dx=1 $$