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# Probability of two consecutive events

Mon 18 Sep, 2023 04:59 am
Hello, there!

I'd like to ask for help with a problem I'm working on in my spare time. I have a collection of 20000 points of data, that I managed to categorize as a normal distribution. I'd like to test a hypothesis accross the input values, meaning, given a certain x, we all know the p(x) from the normal distribution, and the farther is x from the average of the distribution, the lower is p(x). But then, what would be the probability of x making a 'return' toward the average in the event that x happens?

It's not going to be p(x)*p(ave), because this product would by definition be lower then p(x), and we know that the farther x is from the average, the higher the likelyhood it will come back towards the average. How can I state this problem mathematically? I don't understand what I have to do. Please, help.

Thank you.
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engineer

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Mon 18 Sep, 2023 06:10 am
@VincentValentine,
To make things easier, I'm going to use the fact that the normal distribution is symmetric around the mean, so we just need to consider values on one side of the distribution.

For any given point x, the probability that it will move towards the mean is 1-Fx(x) where Fx(x) is the cumulative probability function. So you need to integrate p(x) * (1-Fx(x)) from x equals -infinity to the mean. If you are using excel, the NORMDIST function can give you both the normal function and the cumulative normal function.
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