Exercise II: Gaussian integral For K > 0, define RK = [−K, K] × [−K, K], DK = {(x, y) ∈ R 2 , x2 + y 2 ≤ K2 }. 1. Compute Z Z DK e −x 2−y 2 dA. 2. Argue that Z Z DK e −x 2−y 2 dA ≤ Z Z RK e −x 2−y 2 dA ≤ Z Z D√ 2K e −x 2−y 2 dA. 3. Deduce the value of Z +[infinity] −[infinity] e −x 2 dx. Recall that, by definition, Z +[infinity] −[infinity] e −x 2 dx = lim K→+[infinity] Z K −K e −x 2 dx. Be clear and precise and use a famous result to compute limits!

Walgreens sells both kleenex and nyquil. in a random sample of 5000 purchases, it was found that 1200 contained kleenex and 540 contained nyquil. these are not independent because sick people are more likely to buy both and healthy people more likely to buy neither. in looking into that, it was found that 360 purchases contained both. a. make a two-way table to display this information. fill in the table with the missing entries, and be sure your table has row and column totals. b. if we randomly select a purchase that contained kleenex, what is the probability it also contained nyquil? c. what percent of all purchases didn’t contain either product?