4. Let X be the number of major storms in a particular state in a given year. The probability of having no major storms is 0.50, of one major storm is 0.30, of two major storms is 0.10, of three major storms is 0.08, and of four major storms is 0.02 (there have never been more than 4 major storms in this state in a given year). STA 103 Fall 2016 Homework 2 - Due Wednesday, October 12th 1. At a particular veterinarian school, 75% of all students are female. 60% of females play either computer games, console games, or both. 75% of males play either computer games, or console games, or both. Out of those who are female, 30% play computer games, and 10% play both console games and computer games. Out of those who are male, 20% play both computer games and console games, and 15% play console games only. (a) Write down the probability distribution of X in table form, and also check to make sure it is a valid probability distribution. (b) Find the expected number of major storms in a given year, and the standard deviation of the number of storms in a given year. (a) For women only, draw a representation of the sample space, and the numerical information you have. (b) Repeat part (a) for men. (c) Find the probability that there is at least 1 storm in a given year. (c) What is the probability that a female plays only console games? (d) If there was more than 1 storm, find the probability that there were 3 storms. (d) What is the probability that a male plays only computer games? 5. Let X be a continuous random variable defined on the interval [0, 4] with probability density function (e) What is the probability of picking a female, if they play both computer games and console games? p(x) = c(1 4x) (f) Is gender independent of playing computer games? Justify your answer. (a) Find the value of c such that p(x) is a valid probability density function. (b) Find the probability that X is greater than 3. 2. Suppose that start-up companies in the area of biotechnology have probability 0.20 of becoming profitable and that those in area of information technology have probability 0.15 of becoming profitable. Further assume that the start-up companies in biotechnology and information technology are independent. (c) If X is greater than 1, find the probability X is greater than 2. (d) What is the probability that X is less than some number a, assuming 0 < a < 4? 6. Let X be a continuous random variable defined on the interval [1, 10] with probability density function (a) What is the probability that both companies become profitable? p(x) = cx2 = (b) What is the probability that neither company becomes profitable? (a) Find the value of c such that p(x) is a valid probability density function. (c) What is the probability that at least one of the two companies become profitable? (b) Find the probability that X is larger than 8 or less than 2 (this should be one number!). 3. A student offered random people 5 cookies total. Let X be the number of cookies that the person took, and let X have the following probability distribution (you may assume that these are population values): X P (X) 0 0.05 1 0.25 2 0.38 3 0.18 4 0.11 c x2 (c) Find the probability that X is larger than some value a, assuming 1 < a < 10. (d) Find the probability that X is more than 3. 7. Assume that each time that I commute to Davis, the probability I see a police officer on any given day is 0.85. If I commute 7 days a week, solve the following problems: 5 k (a) Find the probability I see a police officer on exactly 4 of the days I commute. (a) Find the value k such that X has a valid probability distribution. (b) Find the probability I see a police officer on at most 5 of the days I commute. (b) Find X . 2 (c) Find X . (c) The expected number of days I see a police officer. (d) What is the probability of three or more cookies being taken? (d) If I see a police officer on 3 of the days I commute, what is the probability that I see 2 more? (e) What is the most likely number of cookies that a random person will take? 8. Suppose that the probability of getting an A in a particular course is 0.08, and assume that the all student grades are independent. If you randomly sample 20 students taking the course; (f) If you know that someone took less than 3 cookies, what is the probability they took 1 cookie? 1 (a) Find the expected number of students that will get an A, and the standard deviation of number of students that will get an A. (b) Find the probability that no student gets an A. (c) Find the probability that at most 2 students get an A. (d) Find the probability that between 2 and 4 students get an A (inclusive). (e) If (in a different course), the probability that no students out of 20 got an A was 0.1000, what was the probability that a single student got an A? You may assume that all students were independent, and the probability of an A does not change. 2