Let {W(t) : 0 = t = 1} be a Wiener process with W(0) = 0 and the parameter 2 = 1 (such a process is called a standard Wiener process).

Let {W(t) : 0 <= t <= 1} be a Wiener process with W(0) = 0 and the parameter α2= 1 (such a process is called a standard Wiener process).

When answering the following questions be sure to justify your answers.

a.) Let X(t) = W(t +α) - W(t) for some α > 0. Is {X(t)} a Gaussian Process?

b.) Is {X(t)} a Wiener process?

c.) Is {X(t)} mean-square continuous?

d.) Let B(t) = W(t) - t*W(1) for 0 <= t <= 1. Is {B(t)} a Gaussian Process?

e.) Derive the covariance function for {B(t)}. 

f.) Show that for any t, B(t) is independent of W(1).