\begin{align*} F(x) & = P(X \le x)\\ & = 1 - P(X \gt x)\\ & = 1 - P(\text{fewer than r successes in [0,x]})\\ & = 1 - \big [ \frac{(\lambda x)^0 e^{-\lambda x}}{0!} + \frac{(\lambda x)^1 e^{-\lambda x}}{1!} + ... + \frac{(\lambda x)^{r-1} e^{-\lambda x}}{(r-1)!} \big ]\\ & = 1 - \sum_{k=0}^{r-1} \frac{(\lambda x)^k e^{-\lambda x}}{k!} \end{align*}