# Probability at different time scales

• garethtilley
In summary, the conversation revolves around finding a general summary for a single random variable, X, based on multiple distributions generated at different time scales. The goal is to take into account the order of the times and find a way to summarize the probabilities of X remaining constant for various times into the future. One proposed solution is to build a bivariate distribution on time and value, but the challenge lies in summarizing the probabilities in a meaningful way.
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

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 modeled as q(x)r(t).

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.

## 1. What is probability at different time scales?

Probability at different time scales refers to the study of the likelihood of events occurring over different periods of time. This can range from short-term events, such as flipping a coin, to long-term events, such as predicting the stock market's performance over the next year.

## 2. How is probability at different time scales calculated?

The calculation of probability at different time scales is based on the principles of probability theory, which uses mathematical formulas to determine the likelihood of an event occurring. The calculation takes into account various factors, such as the number of possible outcomes, the frequency of each outcome, and any relevant data or information.

## 3. What are some real-world applications of probability at different time scales?

Probability at different time scales has many applications in various fields, including finance, economics, weather forecasting, and sports analytics. For example, probability can be used to determine the likelihood of a certain stock increasing in value over a specific period of time, or the chances of a sports team winning a championship in a given season.

## 4. How does probability at different time scales impact decision-making?

Understanding probability at different time scales can help individuals and organizations make more informed decisions. By analyzing the likelihood of different outcomes over various time periods, decision-makers can better assess risks and potential outcomes, and make more strategic choices.

## 5. What are some common misconceptions about probability at different time scales?

One common misconception is that probability only applies to short-term events and cannot be used for long-term predictions. In reality, probability can be applied to events of any duration. Another misconception is that probability can accurately predict the future, when in fact it is based on the likelihood of events occurring and cannot guarantee a specific outcome.

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