Question 3 (10 marks, 5 each) (a) The distribution of a waiting time for an event to occur is often modeled by an exponential distribution p(yle) = de #, y>0 with 0 > 0 being the parameter to be estimated. Given a dataset y = {y1, ...,y,} with the sample mean y = 10, find the maximum likelihood estimate of 0. (b) A company will introduce a new product if at least do X 100% of its customers purchase the product, do has been determined. Let 0 be the probability that a customer would buy the product. One needs to estimate 0, then compare it with do. Suppose that n customers * = (@1, ..., In) are surveyed, with r, = 1 meaning "Yes", and I; = 0 meaning "No". Statistically, we say that each x; has a Bernoulli distribution B(1, 0) with density p(x: |0) = 0=(1 - 0)1-zi Write down the likelihood function based on sample x. What is MLE of 0