I am modeling three events A, B, and C as Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$ and I would like to calculate the likelihood of observing some data given my model.
A distinction that I have come across is the difference between an event happening at a specific time $t$, $P\left(T\left(E\right) = t \right)$, versus by time $t$, $P\left(T\left(E\right) \leq t \right)$, where $T\left(E\right) = E_n – E_{n-1}$ refers to the inter-arrival time between two consecutive occurrences of the event. For a single event with rate $\lambda$, the probability of these scenarios are the exponential likelihood function
$P\left(T\left(E\right) = t \right) = \lambda e^{-\lambda t}$
and an integration over it
$P\left(T\left(E\right) \leq t \right) = \int_0^t \lambda e^{-\lambda \tau} \, \textrm{d}\tau = 1 – e^{-\lambda t}$
of the exponential distribution, respectively.
It follows that the odds of the event not occurring by $t$ is
$P\left(T\left(E\right)\gt t\right) = 1 – \left(1 – e^{-\lambda t}\right) = e^{-\lambda t}$.
Returning to my three-event model, the probability of observing event A at time $t$ but not events B nor C by that time is
$P\left(T\left(A\right) = t \cap T\left(B\right) \gt t \cap T\left(C\right) \gt t \right) = \lambda_A e^{-\lambda_A t} e^{-\lambda_B t} e^{-\lambda_C t} = \lambda_A e^{-\lambda t}$
where $\lambda = \lambda_A + \lambda_B + \lambda_C$, the overall rate of events.
However, I see two non-equivalent approaches to calculate the probability of observing event A by time $t$ but not events B nor C by that time, $P\left(T\left(A\right) \leq t \cap T\left(B\right) \gt t \cap T\left(C\right) \gt t \right)$.
The first method I see is to explicitly evaluate each expression, as I did previously; that is,
$P\left(T\left(A\right) \leq t \cap T\left(B\right) \gt t \cap T\left(C\right) \gt t \right) = \left(1 – e^{-\lambda_A t}\right) e^{-\lambda_B t} e^{-\lambda_C t} = e^{-(\lambda-\lambda_A)t} – e^{-\lambda t}$.
The second is to integrate over all possible times:
$P\left(T\left(A\right) \leq t \cap T\left(B\right) \gt t \cap T\left(C\right) \gt t \right) = \int_0^t \lambda_A e^{-\lambda \tau} \, \textrm{d}\tau = \frac{\lambda_A – \lambda_A e^{-\lambda t}}{\lambda}$.
Both approaches seem equally valid to me. Which calculation should I be using, if either, and why? Thanks in advance for your help; I really appreciate it.
Best Answer
First of all, I can't see how the random variable $T(E)$ has $P(T(E)=t)$ non-zero, assuming continues time.
The first method I see nothing wrong with assuming these events are independent. Are they?
I can't see how you got the integral for the second method.