1.An exam consists of n questions. Before the exam, m teachers are going to beta test the exam. Each of the teachers will solve a question correctly with the probability p (independently of other teachers and independently of other questions). Assume X be the number of distinct questions that no one solves correctly.
(1) What is the expectation of X (i.e, E[X]) ? What is the variance of X ?
(2) Now each teacher is going to choose a question uniformly at random from the n questions to grade (independently of other TAs). Assume Y be the number of distinct questions that no one chooses. What is the expectation of Y (i.e, E[Y]) ? What is the variance of Y ?
2.A random variable X is always strictly larger than -100. We have E[x] = -60. Give the best upper bound you can on P(X>= -20).