False Discovery in Alpha Mining

You are mining for trading alphas. You test $M$ independent strategies, each producing a t-statistic $Z_1, \ldots, Z_M \sim N(0,1)$ under the null hypothesis that the strategy has no alpha. You declare a strategy "significant" if $Z_j > z$, where $z = \Phi^{-1}(1 - \alpha)$ is the $(1 - \alpha)$-quantile of the standard normal. 1. Compute the expected number of false discoveries. 2. Compute the probability of at least one false discovery. 3. Choose $\alpha$ (in terms of $M$ and a target $\delta \in (0,1)$) so that the probability of any false discovery is at most $\delta$ (Bonferroni-style control).