1. Let W be a normal (Gaussian) random variable with mean 0 and variance 1. Compute the values of the moments:
![](https://book.transtutors.com/qimg/9e7df41e-08f3-49de-817f-f794e455b651.png)
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 {Wt}t=0,...,N is a Gaussian white noise with variance σ2, and for each α ∈ R, let us define the time series {Xt}t=0,...,N by the formula:
![](https://book.transtutors.com/qimg/81d3e468-9c7c-4de8-b50f-8786d1dc8347.png)
Compute the mean, variance and auto-covariance functions of the time series {Xt}t=0,...,N . Is it stationary?
4. Obviously, Xt strongly depends upon Wt, however, find a value of a for which Wt and Xt are uncorrelated?