Problem 2 (14 points): You are a quality engineer at a clothing manufacturing firm, Granada Moose, in charge of maintaining the production quality of their signature jackets. Despite your best efforts, the production process creates some number of defects in each jacket. Suppose the number of defects X in a jacket is distributed as a Poisson random variable with (unknown) parameter A. (Recall that if X ~ Poisson (A), then P(X = n) = en/n! for each n > 0.) (a) (4 points) Suppose you inspect 5 jackets at random, and observe that they contain the following number of defects: x = (2, 5, 3, 6, 1). Write down the log-likelihood function log L(x). (b) (3 points) For x = (2,5, 3, 6, 1), find the maximum likelihood estimate mle for A. (c) (3 points) After making some much needed changes, you have now improved the production quality so that X is now distributed as a Poisson random variable with parameter >= 1. A jacket meets your firm's quality standards if the number of defects is less than or equal to 2. What proportion of the jackets your firm makes meets its quality standards? (d) (4 points) Suppose your firm ships the jackets in batches of N = 1000 to a retailer. To ensure the quality level of the entire batch, the retailer performs acceptance sampling, where they randomly select 10 jackets from the batch, and count the number of jackets that do not meet the quality standard. The retailer sends back the entire batch of jackets if the number of jackets that do not meet the quality standard is strictly more than 3. What is the risk your firm faces with this procedure? (Since the batch size is large, you may use the binomial approximation.)