Let \(A\) and \(B\) be independent events. Let \(\mathbf{1}_A\) and \(\mathbf{1}_B\) be the associated indicator random variables.
- Describe the random variable \(\mathbf{1}_A+ \mathbf{1}_B \) in terms of \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
- Calculate \(\mathbf{E}(\mathbf{1}_A+ \mathbf{1}_B )\).
- Describe the random variable \((\mathbf{1}_A+ \mathbf{1}_B )^2\) in terms of \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\) ?
- Calculate \(\mathbf{E}\big( (\mathbf{1}_A+ \mathbf{1}_B )^2 \big)\).
[Partially inspired by Pitman p182, #10]
