Problem 3. We have seen the Bonferroni corrections for the simultaneous confidence intervals a couple of times but we never explained why that works. Now let's take a look. (1) Consider two events A and B. Prove that P(An B) 2 P(A) + P(B) - 1. Demonstrate using a Venn diagram will suffice. (2) Consider m events Al, ..., Am. Show that the probability that they happen simultaneously can be bounded as follows: m m Ai 12 P(Ai) - (m - 1). i=1 (3) Suppose we have m confidence intervals for m different parameters. Each confidence interval statement can be construed as an event. Show that to have an overall confidence level of 1-a, each individual confidence interval can simply be adjusted to the level of 1 - a/m