F09 - Further Special Discrete Probability Distributions - Assignment

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FM / Further Special Discrete Probability Distributions Page 1 of 1 National Junior College 2016 – 2017 H2 Further Mathematics Topic: Further Special Discrete Probability Distributions Assignment Name:____________________________________________________ Time Spent: __ h ___ min 1 In a skiing resort, the probability that snow will fall on any day during the winter season is 0.2 on average. The first day of the winter season is 1 December. (a) State two assumptions required for the number of days from 1 December for snow to first fall to be well-modelled by a geometric distribution. [2] Assume for the remainder of the question that the assumptions you have stated in part (a) hold. (b) Find, for the winter season, (i) the probability that the first snow falls on 20 December, [1] (ii) the probability that the first snow falls before 5 December, [1] (iii) the probability that the first snow falls after 20 December, if it has not snowed yet and it is now 5 December. [2] (iii) the earliest date in December such that the probability that the first snow falls on or before that date is at least 0.95. [3] 2 The number of radioactive particles emitted per 150-minute period by some material has a mean of 0.7. (a) State two assumptions required for the number of radioactive particles emitted by the material in any hour to be well-modelled by a Poisson distribution. [2] Assume for the remainder of the question that the assumptions you have stated in part (a) hold. (b) Find (i) the probability that more than 2 particles will be emitted during a randomly chosen 10 hour period. [2] (ii) the probability that out of three randomly chosen 10 hour periods, more than 2 particles will be emitted in each of two of these 10 hour periods. [2] (iii) the longest time period, in minutes, for which the probability that no particles are emitted is at least 0.99. [4] (c) There is a probability of 0.324135 that 4 particles are emitted during the first n hours of a 20 hour period, given that 6 particles are emitted in total in this 20 hour period. Set up a polynomial equation in n, and show that there is exactly one integer solution for n. [6] 3 Show that if f is a geometric distribution function, then the function g : [0,1]  defined by 2 f (1)g( ) f (2 )1 f (1)x x    is a probability distribution function. [5] Find the mean and variance of a random variable with probability distribution function g in terms of p, where p = f(1). [9] www.KiasuExamPaper.com 822