MACT 3224/317: Assignment N 2 Yahia El Horbaty, Khouzeima Moutanabbir Abd-Elnasser Saad and Noha Youssef Fall 2016 Return this page with your signature Name: Student ID: Section number: I certify that this is my own work, and that I have not copied the work of another student. Date: Signature: Please follow the instructions below Write your name and section number at the spaces provided. Give clear answers and explanations justifying your method wherever possible. Due Thursday, October 20, 2016 by 2:00 pm. Submit your work to the teaching assistant. Homework cannot be submitted as an e-mail attachment. Late submissions are not allowed. The Department of Mathematics and Actuarial Science, The American University in Cairo. 1 Question 1: Consider a random variable X with the following moment generating function mX (t) = 0.2e2t , 1 0.8e2t Find (a) Find the expected value of 3 X (b) Determine the variance of 3 X Question 2: Given that Y has a moment generating function mY (t) = \u00012 1 2t e +3 , 16 for t 0. (a) Find P (y) the probability distribution of Y (b) Determine the mean and the variance of Y . Question 3: A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a 3D television set, and 30 percent will just be browsing. If ten customers enter his store on a certain day, find the probabilities of the following events: (a) two customers purchase 3D sets; (b) one customer purchases an ordinary set; (c) two customers purchase nothing. Question 4: The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the probability mass function of X is given as follows: x P r(X = x) 24 0.0825 25 k 26 0.7372 27 0.0541 28 0.0518 (a) Find k. (b) Determine the expected number of neutrons in a randomly chosen atom. Question 5 The probability that a computer running a certain operating system crashes on any given day is 0.1. (a) Find the probability that the computer crashes for the first time on the 14th day after the operating system is installed. (b) Find the probability that the computer crashes for the third time on the 34th day after the operating system is installed. (c) Find to probability that the computer crashes 3 times in the first 34 days after the operating system is installed. Question 6 Microprocessing chips are randomly sampled one by one from a large population and tested to determine if they are acceptable for a certain application. Ninety percent of the chips in the population are acceptable. We define the random variables 2 - X: the number of chips that are tested up to and including the first acceptable chip; - Y : the the number of chips tested up to and including the second acceptable chip. Calculate (a) P r (Y = 4 | X = 2) (b) P r (X = 2 | Y = 4) Question 7 Particles are registered by a counter at an average rate of particles per second and this number is distributed following Poisson distribution. It is known that the probability to register two particles is twice the probability to register three particles. (a) Find the value of . (b) Find the probability that at most two particles are registered in a given second. (c) Find the probability that at least three particles are registered in two seconds. Question 8 A box contains 7 marbles of which 3 are red and 4 are blue. Randomly select two marbles without replacement. If the marbles are of the same color then you win $2, otherwise you lose $1. Let X be the random variable representing your net winnings. (a) Find the possible values of X and their probabilities. (b) Compute the expected gain E(X). \u0002 \u0003 (c) Compute E 2X . Question 9 An insurance policy pays an individual 100 per day for up to 3 days of hospitalization and 25 per day for each day of hospitalization thereafter. The number of days of hospitalization, X, is a discrete random variable with the following probability function P (X = k) = 6k , 15 for k = 1, 2, 3, 4, 5. (a) Determine the possible values of the payment for hospitalization under this policy. (b) Calculate the expected payment for hospitalization under this policy. 3