$Operational Research probabilistic models$

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$Operational Research probabilistic models$

Recall the following definition of a Poisson process with rate λ > 0 : a counting process {N(t)} with N(0) = 0 is called a Poisson process with rate λ if the following are true: it has independent increments; and for each t > 0, N(t) is a Poisson random variable with parameter λt. Using the above definition to show that, for each v, t > 0, N(t + v) − N(t) follows Poisson distribution with parameter λv. [20 marks]

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Due to independent increments we have

\[P(N(t+v)-N(t)=n)= \text{Probability of one $n$ occurrences between in } [t, t+v]\]
\[=\frac{(\lambda v)^ne^{-\lambda v}}{n!},\]
the above equality follows from the assumption that $N(t)$ is a Poisson random variable with parameter $λt$, and it has independent increments. Hence
\[P(N(t+v)-N(t)=n)=\frac{(\lambda v)^ne^{-\lambda v}}{n!},\]
which means that $N(t+v)-N(t)$ is a Poisson distribution with parameter $\lambda v$.

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I meant " Probability of n occurrences ... "

$Operational Research probabilistic models$

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