<p>I assume that $r(t)$ is continuous.</p>

<p>The idea is that, this Poisson process with time-varying parameter $r(t)$, as the limit of Bernoulli trial with time-varying probability of success, is memoryless:</p>

<p>For a generic finite partition $\mathscr{P}$ of $[0, T]$ as $\{[0,t_1),[t_1,t_2),[t_2,t_3) \ldots,[t_{n-1},T]\}$, let the $n$th cell following a Poisson distribution with a fixed parameter $r(t_n)$, so 
 $$P_{\mathscr{P}} = 1 - \underbrace{e^{\sum_{k=1}^n-r(t_k)(t_k-t_{k-1})}}_{\text{Probability of no accidents}}$$. So  $P = 1 - e^{-\int_0^Tr(t)dt}$. Notice that the Riemann integral $\int_0^Tr(t)dt$ is the limit of a net indexed by the set of all finite partitions of $[0,T]$ with vanishing maximal length of cell with respect to number of cells. $f(x)=1-e^{-x}$ is continuous, so this limit preserves.</p>

CC BY-SA 3.0
2015-07-09T16:47:13.683Z
2015-07-09T21:54:07.817Z
2015-07-09T21:54:07.817Z
Metta World Peace
<p>We live in continuous time $t$ and something terrible is happening at a poisson rate of $r(t)$.</p>

<p>How can I compute the length $T$ such that with a probability of $P$ (for example, 0.99), at least one terrible thing has happened?</p>

<p>I know that for a constant rate $r(t) = \bar r$, the expected value is $1/\bar r$. How can I go on?</p>

<probability>
Something Terrible is Happening - But When Is it Likely to have Happened?
FooBar
# Something Terrible is Happening - But When Is it Likely to have Happened?

(Asked by: username_0 on 2015-07-09)

We live in continuous time $t$ and something terrible is happening at a poisson rate of $r(t)$.

How can I compute the length $T$ such that with a probability of $P$ (for example, 0.99), at least one terrible thing has happened?

I know that for a constant rate $r(t) = \bar r$, the expected value is $1/\bar r$. How can I go on?

**Tags:** probability

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## 1 answers

### Answer by username_1 on 2015-07-09. Score: 3 | Accepted answer

I assume that $r(t)$ is continuous.

The idea is that, this Poisson process with time-varying parameter $r(t)$, as the limit of Bernoulli trial with time-varying probability of success, is memoryless:

For a generic finite partition $\mathscr{P}$ of $[0, T]$ as $\{[0,t\_1),[t\_1,t\_2),[t\_2,t\_3) \ldots,[t\_{n-1},T]\}$, let the $n$th cell following a Poisson distribution with a fixed parameter $r(t\_n)$, so
$$P\_{\mathscr{P}} = 1 - \underbrace{e^{\sum\_{k=1}^n-r(t\_k)(t\_k-t\_{k-1})}}\_{\text{Probability of no accidents}}$$. So $P = 1 - e^{-\int\_0^Tr(t)dt}$. Notice that the Riemann integral $\int\_0^Tr(t)dt$ is the limit of a net indexed by the set of all finite partitions of $[0,T]$ with vanishing maximal length of cell with respect to number of cells. $f(x)=1-e^{-x}$ is continuous, so this limit preserves.


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