Suppose that liquid in a container is placed in a coordinate system, and at time 0, a pollen particle suspended in the liquid is at (0, 0, 0), the origin. Let Z(t) be the z-coordinate of the position...

Suppose that liquid in a container is placed in a coordinate system, and at time 0, a pollen particle suspended in the liquid is at (0, 0, 0), the origin. Let Z(t) be the z-coordinate of the position of the pollen after t minutes. Suppose that $ Z(t): t ≥ 0 % is a Brownian motion with variance parameter 9. Suppose that after 5 minutes the z-coordinate of the pollen’s position is 0 again.

(a) What is the probability that after 10 minutes it is between −1/2 and 1/2?

(b) If after seven minutes the z-coordinate of the pollen’s position is −1, find the expected value and variance of the z-coordinate of the position of pollen after six minutes.