I was reading about the geometric distribution, where $E(X) = 1/p$. This made me think of the classic formula in physics: $t=1/f$. Is there a parallel between both formulas, especially since a probability is often viewed as a frequency, and, on the other side of both equations, I have a time?
Assume we do a Bernoulli trial every second with a probability $p=0.01$. If we assign 1 to a success and zero to a failure, we can create a random time-series, or a signal where the long-term average frequency should be 0.01 Hz. The long-term average period of this signal, $t$ should be $1/f = 100$ seconds.
- I can expect, on average, to see a full cycle ($t$) every 100 seconds if $f$=0.01Hz.
- I can expect, on average, my first success after 100 seconds: $E(X)$ of a geometric distribution with p = 0.01, and one draw per second.
Is the resemblance in the two formulas a coincidence, and if so, where is the flaw in the logic? Or is there a parallel across physics and statistics? And if so, how could this be explained more rigorously?
