Finding PDF of a random variable From its CDF

Question

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...

Answer 1

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

Answer 2

You should keep in mind that CDF must require following properties: $$F(x) \to 1 ~ as ~ x \to \infty$$ $$F(x) \to 0 ~ as ~ x \to -\infty$$ $F(x)$ is right-continuous function.

In particular, you want to use first two properties to get rid of indefiniteness while finding CDF as antiderivative of PDF.

As far as finding PDF in this example, I don’t quite get what the problem is here. Everything you need to do to get PDF is to directly calculate derivative for every of the four parts in this system.