Exponent probability density function would be defined as:
$$ f(x)= \begin{cases} \lambda e^{-\lambda x}, & x \ge 0. \\ 0, & x\le 0. \end{cases}$$
Now if i want cumulative distribution function from this i integrate this from $- \infty$ to $\infty$? this should be:
$$F(x)=\int_{-\infty}^x f(s) \, ds=\begin{cases} 1-e^{-\lambda x}, & x> 0. \\ 0, & x\le 0\end{cases}$$
Now there are few things i dont understand. Why we are integrating with different variable than we originally had ? and why we are using $x$ as upper limit ? shouldn't this be $\infty$.
Also if someone could explain all the steps between defining integration and end result. Since i've been trying integrate this by hand but it doesn't seem i would get correct result.
thanks,
Tuki