Solve the questions in the attachment.

1 The number of claims arising in a month under a home insurance policy follows a Poisson distribution with mean 0.075. Calculate the approximate probability that at least 50 claims in total arise in a month under a group of 500 independent such policies. 2 If X has a gamma distribution with parameters o =10 and 2=0.2, calculate the probability that X exceeds 80 style (a) using a normal distribution (b) using a chi-squared distribution. Explain which of these answers is more accurate. [5] 3 When using the continuity correction with a random variable X that can take any integer value, write down expressions that are equivalent to the following: (i) X 1 ) and 1. (i) (a) Show that the mode of Y is given by: a-1 (b) By considering the relative locations of the mean and mode using sketches of the gamma distribution, state how you would expect the distribution to behave in the limit as o -> , but where 1 is varied so that the mean - has a constant value /. (Wi) Given that o =50 and 1=0.2, calculate the value of P(Y > 350) using: (a) the chi-squared distribution (b) the Central Limit Theorem. (iii) Explain the reason for the difference between the answers obtained in part (ii). 11 (1) X1.....X are independent and identically distributed Gamma(a, 1) random variables. Show, using moment generating functions, that X has a Gamma(na, n)) distribution. (ii) If the random variable 7 , representing the total lifetime of an individual light bulb, has an Exp(4) distribution, where 1/2=2,000 hours, calculate the probability that the average lifetime of 10 bulbs will exceed 4,000 hours