= Solution
The event that the first arrival exceeds $t$ is exactly the event $N_t=0$. Thus $\mathbb P(\tau_1>t)=e^{-\lambda t}$ for $t\geq0$, and differentiation gives the <exponential distribution> density $\boxed{f_{\tau_1}(t)=\lambda e^{-\lambda t}}$ for $t>0$.
Integration by parts, or the gamma integral, yields $\mathbb E\tau_1=\int_0^\infty e^{-\lambda t}dt=1/\lambda$ and $\mathbb E\tau_1^2=2\int_0^\infty te^{-\lambda t}dt=2/\lambda^2$. Therefore \b[$\boxed{\mathbb E\tau_1=1/\lambda,\quad\operatorname{Var}\tau_1=1/\lambda^2}$].
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