Determine as functions of sigma the values of the same

Question

1. Let W be a normal (Gaussian) random variable with mean 0 and variance 1. Compute the values of the moments:

2. Let us now assume that W is normal with mean 0 and variance σ2. Determine (as functions of σ) the values of the same moments as before.

3. Let us assume that {W_t}_t=0,...,N is a Gaussian white noise with variance σ², and for each α ∈ R, let us define the time series {X_t}_t=0,...,N by the formula:

Compute the mean, variance and auto-covariance functions of the time series {X_t}_t=0,...,N . Is it stationary?

4. Obviously, X_t strongly depends upon W_t, however, find a value of a for which W_t and X_t are uncorrelated?