How to integrate a piece-wise function and what are semicolons in function notation? I'm trying to get the cdf from a pdf, which, according to this MIT open courseware video, is as simple as getting integral from -$\infty$ to $\infty$. Like this:

$$
F_z(z) = \int_{-\infty}^\infty f_z(z)dz\, = 1
$$

And that works for functions like this: $f_x(X)$

But what about a function like this $f(X;\beta)$?

Yeah, this is a homework problem, so I don't expect an exact answer, but hopefully some general principles. I have a hard time googling to learn how to do this because I don't know what to call this kind of function $f(X;\beta)$, and I don't know how to type a $\beta$ into google.

The pdf I have is:

$$
f(X;\beta)=
    \begin{cases}
      \frac {e^{-X/\beta}}{\beta}, &amp; \ X\geq0 \\
      0, &amp; \text{otherwise}
    \end{cases}
$$

and the cdf it comes out to is:
$$
f(X;\theta)=1-e^{-X/\beta}
$$

Part of my question is also, why did the $f(X;\beta)$ become $f(X;\theta)$?
CDF = integral of PDF from $-\infty$ to $x$, or:

$F\left(x\right) = \int_{-\infty}^{x} f\left(u\right) du$.

Functions like $f\left(x;\beta\right)$ take $\beta$ as the parameters, the $\beta$ index different probability density functions.  You don't need to integrate over them to get the CDF.

$F\left(x;\beta\right) = \int_{-\infty}^{x} f\left(u;\beta\right)du$.