determine distribution function from density function

Question

The variable $\xi$ has the following density function:

$$f(x)=\begin{cases}x/50&0<x<10\\ 0&else\end{cases}$$

How do I determine its distribution function?

Answer 1

You get the CDF by integrating $f(x)$: $$F(x) = \int_0^xf(s)\;ds = \int_0^x\frac{s}{50}\;ds=\frac{x^2}{100},\qquad 0<x<10$$ You can write $$F(x) = \begin{cases} 0 & x\leq 0\\ \frac{x^2}{100} & 0<x<10 \\ 1 & x \geq 10 \end{cases}$$

Answer 2

The definition of the Cumulative Distribution Function (CDF) follows as :

$$F_X(x) = P\{X\leq x\} = \int_{-\infty}^xf_X(x)dx =\int_{-\infty}^xf(s)ds =\int_{0}^{x}\frac{s}{50}ds$$