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Cumulative probability that random walk variable exceeds given value
*To*: mathgroup at smc.vnet.net
*Subject*: [mg104834] Cumulative probability that random walk variable exceeds given value
*From*: Kelly Jones <kelly.terry.jones at gmail.com>
*Date*: Wed, 11 Nov 2009 04:28:02 -0500 (EST)
How can I use Mathematica to solve this problem?
Let x[t] be a normally-distributed random variable with mean 0 and
standard deviation Sqrt[t].
In other words, x[0] is 0, x[1] follows the standard normal
distribution, x[2] follows the normal distribution with mean 0 and
standard deviation Sqrt[2], etc.
It's easy to compute the probability that x[5] > 2 (for example).
How do I compute the probability that x[t] > 2 for 0 <= t <= 5.
In other words, the probablity that x[t] surpassed 2 at some point
between t=0 and t=5, even though x[5] may be less than 2 itself. Notes:
% My goal: predicting whether a continuous random walk will exceed a
given value in a given period of time.
% I realize that saying things like "x[5] may be less than 2" is
sloppy, since x[5] is a distribution, not a value. Hopefully, my
meaning is clear.
% I tried doing this by adding/integrating probabilities like this
(psuedo-code):
P(x[t] > 2 for 0 <= t <= 5) = Integral[P(x[t] > 2),{t,0,5}]
but this overcounts if x[t] > 2 for multiple values of t.
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