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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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