Question 2 A discrete random variable X can take only the values, 1, 2 3, or 4 and its probability distribution is given as below, where & is a constant. X 1 2 3 4 P(X=x) & 2k 31 4k Find: (1) the value of A. (2 marks) Substitute the value of & into the table, hence find: (1i) E(3X - 1). (2 marks) (b) The number of passengers on the Carnival Sensation during one-week cruises in the Caribbean follows on the normal distribution. The mean number of passengers per cruise is 1820 and the standard deviation is 120. (i) What is the probability that the cruise will have between 1820 and 1970 passengers? (2 marks) (i1) What is the probability that the cruise will have 1970 or more passengers? (2 marks) (iii) The caption of the top 25% crowded cruise will receive a bonus. What are the minimum passengers for a cruise to be included? Find the number of passengers. (2 marks) (C) A student majoring in accounting is trying to decide on the number of firms to which she should apply. Given her work experience, grades, and extracurricular activities, she has been told by placement counselor that she can expect to receive a job offer from 80% of the firms to which she applies. Wanting to save time, the student applies to only five firms. Assuming the counselor's estimate is correct. (i) Find the probability that the student receives no offers. (2 marks) (i1) Find the probability that the student receives at most two offers. (3 marks) iii) Find the probability that the student receives at least three offers. (3 marks) (iv) Find the average and standard deviation the student will receives an offer. (2 marks) (Total: 20 marks)