# Probability at different time scales

garethtilley
Hi,

Was wondering if anyone had any thoughts on how to tackle the below problem:

I have a single random variable, X. I have generated multiple distributions (not fitted yet, not sure I will/can fit them) at different time scales. So, for example, a distribution of values of X after 10 seconds from time zero and another after 20 seconds from time zero etc. There are no assumptions around mean reversion or unbounded variance or anything like that. Only empirical results. Having generated the various distributions I can at any point do a lookup on them, for some value of X, let's say the current value, and I'll get back a series of probabilities of X remaining constant (in this case) for various times into the future.

So here's my problem, given these multiple time scales, I'd like to try get a generalised summary of sorts of what is going on with X. I've considered a few ideas, like just taking a simple average, but obviously that doesn't take order of the times into account. For example, p10sec = 0.2 and p20sec = 0.8 paint a very different view to p10sec = 0.9 and p20sec = 0.2. Any ideas?

I hope that explanation is clear, I'm no statistician!

Regards
Gareth

Eynstone
Use a distribution function p(x,t) depending on both x & time for X.
If the variable is 'oblivious' (i.e., the distibution which would be observed at a time doesn't affect that at a while later) , p(x,t) can be modelled as q(x)r(t).

garethtilley
So if I understand you correctly, you're proposing a building a bivariate distribution on time and value.

I've considered this, if I built that surface, for a given time I'd need to look into the "future" at all the nodes, i.e. 10s, 20s etc. and I'd still end up with a series of probabilities, that some how still need to be summarised, with the order that they're in being important.

I hope that makes sense and I hope I understood your response.