determine distribution function from density function 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?

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
$$F(x) = \begin{cases}
0 & x\leq 0\\
\frac{x^2}{100} & 0<x<10 \\
1 & x \geq 10
\end{cases}$$
