Answer all the questions.

2010 have the following history: 48 years with 0 vacancies, 23 years with 1 vacancy, 7 years with 2 vacancies, and 0 years with 2 3 vacancies. Test the hypothesis that the number of vacancies per year follows a Poisson distribu- tion.2B.7 Distributions related to the normal Random variables X 1, X2, . . . . Xjop are all independent and all have the normal distribution with mean 0 and standard deviation 1. Write down functions of the X's which have the distributions below, justifying the results you use. (a) N(0, 4) (b) xfo (c) 150 (d) F 48,502A.7 The telephone exchange A telephone exchange receives, on average, 5 calls per minute. Find the probability (i) that in a 1-minute period no calls are received; (ii) that in a 2-minute period fewer than 4 calls are received; (iii) that in a 20-minute period no more than 102 calls are received; (iv) that out of five separate 1-minute periods there are exactly four in which 2 or more calls are received.2A.2 Sampling focoming batches Show that the binomial distribution with index a and parameter p has mean ap and variance mp (1 - p). A company taking delivery of a large batch of manufactured articles accepts the batch if either (a) a random sample of 6 articles from the batch contains not more than one defective article, or (b) a random sample of 6 contains two defective articles, and a second random sample of 6 is taken, and found to contain no defectives. If 20% of the articles in the batch are actually defective, what is the probability that the company will accept the delivered batch?2A.1 The squash match Paul and Eric are playing squash, and Paul is determined to win at least two games. Unfortunately his chance of winning any one game is only }, and this chance remains constant however many games he plays against Eric. The players agree to play 5 games and, if Paul has won at least two by then, play ceases. Otherwise Paul persuades Eric to play a further 5 games with him. What is the probability (i) that only 5 games are played, and Paul wins at least two of them; (ii) that 10 games have to be played, and Paul wins at least two?(b) An insect breeding experiment was conducted in two sections, in each of which 100 insects of a particular species were raised. In the first section a proportion p was expected to have a certain colour variation and in the second section the proportion with this colour variation was expected to be p', but the value of p was not known. In the event there were 22 insects in the first section, and 7 in the second section, which possessed the colour variation. Find the value of p which maximises the probability of this result