Historic data of a computer network suggest that connections with this network follow a Poisson's distribution with an average of 5 connections per minute. Find $t_0$, which has probability equal to $0.9$ to happen at least once before $t_0$
I made an approximation using $\lambda=12$, since $5/60 =1/12$, but my answer doesn't match with the book (book says it's ~$27.5$ seconds, and mine ~$17.0$ seconds)
Could someone help me?
Let $(X_t)_{t \in [0,\infty)}$ be a Poisson process where $t$ is minutes. $$\begin{aligned}P(X_{t_0}\geq 1)&=0.9\\ 1-P(X_{t_0}=0)&=0.9\\ 1-\frac{e^{-\lambda t_0}(\lambda t_0)^0}{0!}&=0.9\\ e^{-\lambda t_0}&=0.1\\ t_0&=-\ln(0.1)/\lambda \end{aligned}$$ If we express $t_0$ in seconds we get $t_0\cdot 60\approx 27.63$ where $\lambda =5$.