Let \(A\) and \(B\) be independent events. Let \(\mathbf{1}_A\) and \(\mathbf{1}_B\)  be the associated indicator random variables.

1. Describe the random variable \(\mathbf{1}_A+ \mathbf{1}_B \) in terms of  \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
2. Calculate \(\mathbf{E}(\mathbf{1}_A+ \mathbf{1}_B )\).
3. Describe the random variable \((\mathbf{1}_A+ \mathbf{1}_B )^2\) in terms of  \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
4. Calculate \(\mathbf{E}\big( (\mathbf{1}_A+ \mathbf{1}_B )^2 \big)\).

[Partially inspired by Pitman p182, #10]