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Results tagged with poisson-process
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user 77222
For questions about the theory or applications of the Poisson process, one of the most widely applied point processes in statistics and elsewhere.
2
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Accepted
Expected waiting time until no event in $t$ years for a poisson process with rate $\lambda$
Let $T_1,T_2,\dots$ denote the exponentially distributed inter-arrival times. Each of these exceeds $t$ with probability $p=e^{-\lambda t}$. Hence the number of occurences $X$ before (but not includ …
- 11.7k
6
votes
Accepted
Poisson Distribution: Estimating rate parameter and the interval length
Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variable …
- 11.7k
5
votes
Conditional distribution of arrival times in Poisson process
Since the inter-arrival times are independent exponentially distributed, the joint pdf of the $n$ first arrival times is
\begin{align}
f(t_1,\dots,t_n)
&=f(t_1)f(t_2|t_1)\dots f(t_n|t_{n-1})
\\&=\la …
- 11.7k