User:AndreasWittenstein/sandbox

$$ \Pr(\mathcal{F}^{\prime}=f^{\prime})=c^{f^{\prime}-1}\cdot(1-c),\quad f^{\prime}\in{\mathbb{N}_1} $$

$$ \Pr(\sum_{\rho=0}^{r-1}\mathcal{F^{\prime}}_\rho=f^{\prime})=\binom{f^{\prime}-1}{f^{\prime}-r}\cdot c^{f^{\prime}-r}\cdot(1-c)^r,\quad f^{\prime}\in{\mathbb{N}_1} $$

$$ \Pr(\sum_{\rho=0}^{r-1}\mathcal{F^{\prime}}_\rho\le f^{\prime})=\sum_{\phi^{\prime}=r}^{f^{\prime}}\binom{\phi^{\prime}-1}{\phi^{\prime}-r}\cdot c^{\phi^{\prime}-r}\cdot(1-c)^r $$