An observer is given a box with a light on top, and given no information about it. At time t0, the light on the box turns on. At time tx, the light is still on.
At time tx, what information can the observer be said to have about the probability distribution of the duration of time that the light turns on? Obviously the observer has some information, but how is it best quantified?
For instance, the observer wishes to guess when the light will turn off, or find the best approximation of E(X | X > tx-t0), where X ~ duration of light being on. This is guaranteed to be a very uninformed guess, but some guess is possible, right?
The observer can establish a CDF of the probability of the light turning off at time t; for t ⇐ tx, p=0. For t > tx, 0 < p < 1, assuming that the observer can never be certain that the light will ever turn off. What goes on in between is the interesting part, and I haven’t the faintest idea how to justify any particular shape for the CDF.
An observer is given a box with a light on top, and given no information about it. At time t0, the light on the box turns on. At time tx, the light is still on.
At time tx, what information can the observer be said to have about the probability distribution of the duration of time that the light turns on? Obviously the observer has some information, but how is it best quantified?
For instance, the observer wishes to guess when the light will turn off, or find the best approximation of E(X | X > tx-t0), where X ~ duration of light being on. This is guaranteed to be a very uninformed guess, but some guess is possible, right?
The observer can establish a CDF of the probability of the light turning off at time t; for t ⇐ tx, p=0. For t > tx, 0 < p < 1, assuming that the observer can never be certain that the light will ever turn off. What goes on in between is the interesting part, and I haven’t the faintest idea how to justify any particular shape for the CDF.