When the motion of a microscopic particle in a liquid or a gas id observed, it i

Question

When the motion of a microscopic particle in a liquid or a gas id observed, it is seen that the motion is irregular because the particle collides frequently with other articles. The probability model for this motion, which is called Brownian motion, is as follows: A coordinate system is chosen at in the liquid or gas. Suppose that the particle is at the origin of this coordinate system at time t=0, and let (X,Y,Z) denote the coordinates of the particle at any time t >0. The random variables X, Y, and Z are i.i.d., and each of them has the normal distribution with mean 0 and variance (Sigma^2)(t). Find the probability that at time t = 2 the particle will lie within a sphere whose center is at the origin and whose radius is 4(sigma).

Explanation / Answer

Let s=sigma. The square of the distance from the particle to the origin is,

D2 = X^2 + Y^2 + Z^2

Thus,

Chi3 = D2/(t s^2) has a Chi-square dist. with 3 d.f.

at t=2,

P[D2 < (4 s)^2] = P[ Chi3 < 8 ] = .934

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