I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function g(t)=f(0,t) instead because $f(t_1, t_3) = f(t_1, t_2) + f(t_2, t_3)$ and consequently $f(s, t) = g(t) - g(s)$
The PDF for $f(t,t+l)$ for a random time t is the Poisson distribution PDF with an interval length l (by Poisson PDF I mean the Poisson PMF formula with real values). Formally:
$$ X_l(t) = f(t,t + l) = g(t+1) - g(t) $$ $$ Pr(X_l \leq x) = CDF_{Pois}(x; \lambda = r \cdot l) = \frac{\Gamma(x+1, r \cdot l)}{\Gamma(x+ 1)} $$ where:
- $l=intervalLength$
- $r=eventRate$
- $\Gamma(x)$ is the gamma function.
- $\Gamma(x,y)$ is the incomplete gamma function.
I also can't figure out how to compute $f$ from this, could I add anything to better define it? It seems like I could compute $f$ if I had a function for $g'$ that would to randomly go up and down while hovering above 0. $g''$ would also have to randomly go up and down. So we don't have to keep defining new functions with random dips and peaks perhaps it would hold that $g'(x) = g''(A_1x + A_2)\cdot B_1 + B_2$. In essence, maybe $g''$ is just $g'$ shifted and stretched, similar to $e^x$.
I'm using $f$ for this SO question.