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Properties of sum of Poisson processes

For 1. we have $$ \mathbb P(N_1(3)-N_1(1)=2) = \mathbb P(N_1(2)=2) = e^{-2\lambda_1}\frac{(2\lambda_1)^2}{2!} = e^{-2}\frac 42 =2e^{-2}. $$ For 2. we have \begin{align} \mathbb P(N_2(5)=3\mid N(5)=5) &...
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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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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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Is every mixed Poisson distribution a compound Poisson distribution, and vice versa?

I think that Puri and Goldie (1979) at https://www.jstor.org/stable/3213382 gives a partial answer to Q1: not every mixed Poisson distribution is a compound Poisson distribution. Puri and Goldie give ...
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