Cumulative probabilities/normal distribution/series of trials
Assume a normal distribution. I understand how to calculate the probability of something being X standard deviations above the mean, for example, mean is 10, standard deviation is 5, probability of >15 is (1-.67)/2.
Furthermore, I understand that if we have say 100 trails, the new mean should be 100*the old mean, new variance is 100X the old variance (square result to get st dev). And it is similarly straightforward to determine the probability of being X st devs above the mean (works just like the above example).
But here is my question, how do I determine the probability of the result being > x st devs above the mean at any point in the 100 trial run, not just at the end?
I mean I could calculate the probability for 1 run, 2 runs, 3 runs, etc up to 100, but I see no way to derive the cumulative probability that it happens at any point during that run.
Any ideas?
Let me know if you need any clarification of my question.
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