       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 is 0, x follows the standard normal
distribution, x follows the normal distribution with mean 0 and
standard deviation Sqrt, etc.

It's easy to compute the probability that x > 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 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 may be less than 2" is
sloppy, since x 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.

--
We're just a Bunch Of Regular Guys, a collective group that's trying
to understand and assimilate technology. We feel that resistance to
new ideas and technology is unwise and ultimately futile.

```

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