Let \(\omega_i\) , \(i=1,\cdots\) be a collection of mutually independent, uniform on \([0,1]\) random variables. Define

\[\eta_i(\omega)= \omega_i -\frac12\]

and

\[X_n(\omega) = \sum_{i=1}^n \eta_i(\omega)\,.\]

1. What is \(\mathbf{E}\,X_n\) ?
2. What is \(\mathrm{Var}(X_n)\) ?
3. What is \(\mathbf{E}\,X_{n+k} | X_n \) for \(n, k >0\) ?
4. What is \(\mathbf{E}(\,X_5^2 \,|\, X_3)\) ?
5. [optional] Write a computer program to simulate some realizations of this process viewing \(n\) as time. Plot some plots of \(n\) vs \(X_n\).
6. [optional] How do you simulations agree with the first two parts ?