Let Z ∼ N (0, 1), and c be a nonnegative constant. Find E(max(Z − c, 0)), in terms of the standard Normal CDF Φ and PDF ϕ. (This kind of calculation often comes up in quantitative finance.)
Hint: Use LOTUS, and handle the max symbol by adjusting the limits of integration appropriately. As a check, make sure that your answer reduces to 1/ √ 2π when c = 0; this must be the case since we show in Chapter 7 that E|Z| = p 2/π, and we have |Z| = max(Z, 0) + max(−Z, 0) so by symmetry E|Z| = E(max(Z, 0)) + E(max(−Z, 0)) = 2E(max(Z, 0)).
I wasn't sure how to start with the maximum bit.