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Integrating Poisson Process w.r.t Time

By Tonelli-Fubini: $$E\bigg[\int_0^tN_sds\bigg]=\int_0^tE[N_s]ds=\lambda\frac{t^2}{2}>0,\,\forall t>0$$ Since for $t>0$ we have $\int_0^tN_sds\geq0$ and $E[\int_0^tN_sds]>0$, then $P(\...
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Swapping Poisson process at fixed time

More generally, if $X(t)$ has rate $\lambda_1$ and $X'(t)$ has rate $\lambda_2$, then $X''(t)$ is a non-homogeneous Poisson process with rate step function $$\lambda(t) = \begin{cases} \lambda_1, &...
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What is the probability that Bond will be injured?

Your reasoning is correct. Let $X_s$ be the number of cars that arrive in $(0,s)$ - this has Poisson distribution with parameter $\frac {\lambda s}{60}$ (with $\lambda = 6$, this reduces to $\frac s{...
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