Actally i have 2 question for you guys. It's not a homework. I just curious.
What is the difference of Erlang distribution and Gamma distribution. On Wikipedia it's said if Erlang is Gamma distribution, that in case $\alpha>0\quad,\alpha \in \mathbb{Z}^+$. Is it mean $\alpha$ on Gamma can be real numbers?
How to get CDF of Chi Square Distribution with Summation (Sigma) Form? I know from Wolfram Alpha that the CDF of Gamma Function on Summation form is the following: $F(x)=\left\{ \begin{aligned} 1- \displaystyle\sum_{k=0}^{n-1} \dfrac{\left(\frac{x}{\beta}\right)^k e^{-\frac{x}{\beta}}}{k!}\quad,x>0 \\ 0\qquad \qquad \quad \text{otherwise} \end{aligned} \right.$
Is it mean that we can replace for $\beta=2$ for CDF of Chi Square?
I don't really understand about this distribution and sorry if i have mistakes on that formula. Thanks and i'll appreciate the answer.
Have you checked the Wikipedia pages for these 3 distributions? All of your questions are answered there.
Yes, the Erlang distribution is a special case of the Gamma distribution where $\alpha$ is an integer; for general Gamma distributions, $\alpha$ can be any positive real.
The chi-squared distribution with $\nu$ degrees of freedom is a special case of the Gamma distribution with parameters $\alpha = \nu/2$ and $\beta = 2$. (Beware, note that Wikipedia uses $\theta$ in place of your $\beta$, and uses $\beta = 1/\theta$ instead.)
The CDF you have written is for an Erlang distribution, not for general gamma distributions. Chi-squared random variables with $\nu$ degrees of freedom where $\nu$ is even are also Erlang random variables, so that CDF holds with $k=\nu/2$ and $\beta = 2$. But otherwise the chi-squared random variable is not Erlang, and you cannot use that CDF in those cases.