Question

10.Hello, I need help with this question, please very explicit and clear answers step by step.

10. Let {Nt} t 0 be a Poisson process with parameter ?. It shows that conditional on a jump occurring in the interval [s, s t], the moment in which the jump occurred is evenly distributed in that interval.

Answer 1

This statement is an important property relating the Poisson process with the uniform distribution. It can be easily proved using conditional probability, and distribution function of the Poisson process itself.

Statement

If exactly one jump of a Poisson process has occurred in [s, s+t], then the time of that occurrence is uniformly distributed in the interval.

Proof

$\textup{Let }\omega \textup{ denote any time increment in }[s,~s+t].\textup{ That is, }\omega \in [0,~t]$

$\textup{Given that }N(t)=1,\textup{ Rate parameter}=\lambda$

$\textup{Let }T_{1}\textup{ denote the time instant when jump occurs. }$

$\textup{For }\omega \leq t,\textup{ the cumulative distribution of }T_{1}\textup{ is obtained as}$

$\mathbb{P}(T_{1}\leq s+\omega|N(t)=1)=\frac{\mathbb{P}(T_{1}\leq s+\omega\bigcap \mathbb N(t)=1)}{\mathbb{P}(N(t)=1)}$

$=\frac{\mathbb{P}(1\textup{ event in }[s,~s+\omega ],~0\textup{ event in }[s+\omega ,~s+t])}{\mathbb{P}(N(t)=1)}$

$=\frac{\lambda \omega e^{-\lambda \omega }\times e^{-\lambda (t-\omega )}}{\lambda te^{-\lambda t}}$

$=\frac{\omega }{ t},~~0\leq \omega \leq t.\textbf{ ~QED.}$