Poisson Fun exercise question Based on your understanding of the Poisson process, determine the numerical values of  $a$  and  $b$  in the following expression.
$$ \int_{t}^∞ {λ^6τ^5e^{−λτ} \over 5!}dτ= \sum_{k = a}^b{(λt)^ke^{-λt} \over k!} $$
Find the following
$a=$ 
$b=$ 
I can't seem to figure out what is a and b in Poisson process!
 A: Poisson processes are typically used to describe a series of event occurrences. Here, we have one with event rate $\lambda$, where the number of events occurring in $t$ seconds is a Poisson random variable with mean $\lambda t$. So, the RHS of the equation can have the following meaning: the probability of number of events occurring in $t$ seconds is between $a$ and $b$, inclusive. 
Poisson process have another useful property: the time between consecutive events are independent and exponentially distributed RVs, with rate $\lambda$ again. These are called inter-arrival times. More importantly, if you sum $n$ of these independent exponential RVs, you get a Gamma RV with $\alpha=n,\beta=\lambda$ which has the meaning "the time required to have $n$ number of events". The LHS of the equation resembles the Gamma distribution very much, with $\alpha=6,\beta=\lambda$, and since the integration is from $t$ to $\infty$, it means the probability of the time required to have $6$ events being larger than $t$, which also means the probability of having 5 or less events in $t$ seconds. 
If you combine these two meanings, what will be $a$ and $b$?
