I'm currently struggling with a problem that sounded rather easy to me at first sight, yet I was not able to obtain a working solution:
I have a random variable that describes how often an event occurs in a given time range $t$. The probability mass function of this random variable is known. Now I need to obtain the probability mass function that describes how often the event would occur in another time range $T$ with $T > t$. For simplicity it would be sufficient to assume that there is a maximum amount of times $N$ the event can occur in any time range.
My idea to solve this problem was:
If I assume the problem to be the other way around, so the probability mass funtion of the larger time range being known and the one of the smaller being unknown, the solution is trivial.
Let $P_t$ and $P_T$ be $N$-dimensional vectors holding the probability mass functions of the event occurring $0, 1, 2, ..., N$ times in the time intervals $t$ or $T$ and $A$ a $N×N$ matrix. Then the problem can be written as the matrix equation $AP_T = P_t$ with the matrix $A$ being triangular and containing the binomial probabilities of having $m$ occurrences in the time interval $t$ when there are $n$ occurrences in the time interval $T$.
$A_{m,n} = 0\text{ if }m > n\text{ else }(n\text{ over }m) (\frac{t}{T})^m (1 - \frac{t}{T})^{n - m}$
Now I wanted to obtain $P_T$ by inverting $A$ and writing the equation as: $P_T = A^{-1}P_t$.
Unfortunately this did'nt work out as the resulting $P_T$ contained both values larger than 1 and negative values which means it is no valid probability mass function. I somehow have the feeling the solution is right in front of my eyes, but I just cannot get there. Searching on the internet also did not bring me closer to the solution.
Thank you very much and best regards.