(See page 50 of the Cryer and Chan text, Exercise 3.2)

To help with parts B and C, think of real world applications of The Random Walk.

A random walk (e_t) is defined as a random number of steps in a random direction. Imagine sitting on a park bench watching one pigeon and its aimless wandering in a patch of grass. It moves 2 steps one direction, turns 45 degrees, moves 1 step, turns 240 degrees, moves half a step, turns 1 degree, moves 5 steps, and so on for an hour. There is no discernable pattern or consistency to the pigeon's travels. So after an hour you can expect the pigeon to be exactly where you first observed the pigeon. It is possible for the pigeon to walk a mile away in that time if it would pick that same direction at every time t, but the probability of that happening is close to nil.

Let Y_t be the position of the pigeon at time t and mu be the origin of the pigeon. The variance of the pigeon's position is going to be correlated to the variance of the pigeon's random walks. With every moment, pigeons 1-5 take a random walk defined by the problem. You can draw 5 circles around the origin to depict the area that the different pigeons could possibly end up in, with the larger areas having the larger variances.

Pigeon #3 (Y_t=mu+e_t) is the simplest of the group, he makes one random walk every time t. It is a random direction and a random distance every moment.

Pigeon #5 (Y_t=mu+e_t-e_t-1) has OCD and it really likes its origin. Every moment, it returns along its previous random walk before performing another random walk. So every moment involves #5 returning to its origin before making another random walk. Obviously, the variance of its position is going to be small as it will always be within one random walk of its origin.

Pigeon #4 (Y_t=mu+e_t-.5e_t-1) is similar to #5, but it only returns half the distance of its previous random walk. Its variance will be larger than #5, but smaller than #3.

Pigeon #1 (Y_t=mu+e_t+e_t-1) wanders the most. After every random walk, it continues in that same direction before adding another random walk every moment. Pigeon #1 has the highest area of variance because it has the highest probability of traveling a large distance from the origin.

Pigeon #2 (Y_t=mu+e_t+.5e_t-1) wanders like #1, but not as far. So its area of wandering will be smaller than #1.

It is still possible for Pigeon #1 to be standing at the origin after an hour while Pigeon #5 is 20 steps away, although a very unlikely possibility.

Once you can imagine the different walk observations, you can describe the difference between #1 and #5 easily. Part C is just asking why #4 and #5 have smaller variances than #1 and #2, because every moment partially reduces the total distance travelled from the origin on an oscillating pattern.