- #1

Capitano

- 2

- 0

- TL;DR Summary
- Probability that one random gaussian event will happen before another one.

For concretness I'll use atoms and photons but this problem is actually just about probabilities.

There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom B described by a gaussian distribution P_B, but centered around T_B with variance V_B. Once they emit one photon, the process stops. What is the probability atom A will emit a photon before atom B? My attempt was something like this:

First, I ask a slightly different question. I start with the probability that A will emit a photon and B will not, between t and t+dt. That should be just

(P_A) x (1-P_B) x dt

since we require that A emits during that interval but not B. Now, my first idea now is to just integrate this expression from -infty to +infty, but I feel that's like demanding that in order for P_A to emit at some time, P_B needs to never emit during the whole time, which is not necessary. Another idea was to integrate up to a time t_f and then integrate over that time to infinity, but I'm not sure about that either

There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom B described by a gaussian distribution P_B, but centered around T_B with variance V_B. Once they emit one photon, the process stops. What is the probability atom A will emit a photon before atom B? My attempt was something like this:

First, I ask a slightly different question. I start with the probability that A will emit a photon and B will not, between t and t+dt. That should be just

(P_A) x (1-P_B) x dt

since we require that A emits during that interval but not B. Now, my first idea now is to just integrate this expression from -infty to +infty, but I feel that's like demanding that in order for P_A to emit at some time, P_B needs to never emit during the whole time, which is not necessary. Another idea was to integrate up to a time t_f and then integrate over that time to infinity, but I'm not sure about that either