Brownian motion: Gaussian distribution

[SoggyBottoms]

Homework Statement

A grain of pollen shows Brownian motion in a solvent, such that the position x(t) on the x-axis varies with time. The displacement during one second, x(t + 1) - x(t), is measured many times and found to have a Gaussian distribution with an average of 0 and standard devation σ. What is the average and standard deviation of the displacement x(t + 100) − x(t) during 100 seconds?

The Attempt at a Solution

I would say the average would still be 0 and the SD still σ. Is that correct?

 

[anathema]

Yes, that is correct. The displacement during 100 seconds would still follow a Gaussian distribution with an average of 0 and a standard deviation of σ. This is because the Brownian motion of the pollen grain is a random process, and the average and standard deviation of the displacement remain constant regardless of the time frame being considered.