Re: Cumulative probability that random walk variable exceeds given

• To: mathgroup at smc.vnet.net
• Subject: [mg104868] Re: Cumulative probability that random walk variable exceeds given
• From: Ray Koopman <koopman at sfu.ca>
• Date: Thu, 12 Nov 2009 06:03:23 -0500 (EST)
• References: <hde054\$sn0\$1@smc.vnet.net>

```Pr[a < x[t] < b] = Erf[ a/Sqrt[2t], b/Sqrt[2t] ]/2,
so integrate from t0 to t1 and divide by (t1-t0):

Assuming[a < b && 0 < t0 < t1, FullSimplify[
Integrate[Erf@@({a,b}/Sqrt[2t])/2,{t,t0,t1}]/(t1-t0)]]

(Sqrt[2/Pi]*((a*Sqrt[t0])/E^(a^2/(2*t0)) -
(b*Sqrt[t0])/E^(b^2/(2*t0)) - (a*Sqrt[t1])/E^(a^2/(2*t1)) +
(b*Sqrt[t1])/E^(b^2/(2*t1))) +
(a^2 + t0)*Erf[a/(Sqrt[2]*Sqrt[t0])] -
(b^2 + t0)*Erf[b/(Sqrt[2]*Sqrt[t0])] -
(a^2 + t1)*Erf[a/(Sqrt[2]*Sqrt[t1])] +
(b^2 + t1)*Erf[b/(Sqrt[2]*Sqrt[t1])])/(2*(t1-t0))

On Nov 11, 1:29 am, Kelly Jones <kelly.terry.jo... at gmail.com> wrote:
> 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.
>
> --
> 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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