deadline for this assignment is 17th april
Probability & Statistics (STAT2003) Assignment 3 The due date/time is given on the Electronic Course Profile. Please use the i-mark system to submit. STAT7003 students have an additional question, marked with a star (*). Ev- eryone may attempt the challenge question (#). This can earn bonus marks, but no help will be given. 1. In a large computing project it was found that there were on average 10 “bugs” per 1000 lines of computer code. Let N be the number of bugs per 1000 lines of, as yet, unchecked code. (a) Explain why N ∼ Poi(10) could be a reasonable model assumption. (b) Assuming that we can use the model under (a), calculate the probability that the new piece of code will have more than 5 errors. 2. We toss two fair dice until their sum is 12. (a) What is probabillity that we have to wait exactly 10 tosses (tosses). (b) What is the probability that we do not have to wait more than 100 tosses. (c) What is the probability that we have to wait more than 100 tosses given that the first 50 tosses did not yield a 12. 3. We select at random five persons from a group of 20 males and 15 females. Let N be the number of males. (a) Determine the distribution of N . (b) Give the expected number of females in the selected group. (c) What is the probability that the group is male-dominated? 4. Two chips are being considered for use in a camera. The lifetime of chip 1 is modelled by a normal distribution with parameters µ = 20, 000 (hours) and σ = 4, 000 (hours). The lifetime distribution of chip 2 is also normal, but with µ = 22, 000 and σ = 1, 000 (hours). Which chip is most likely to reach the target lifetime of the camera of 24,000 hours? 5. A certain transistor has an exponential lifetime with parameter λ = 1/2 (year−1). (a) What is the expected lifetime of the transistor? (b) Suppose after 2 years the transistor is still functioning. What is the probability that it will function for 3 more years? 6. A random variable X has pdf f given by, f(x) = c √ x, x ∈ [1, 9]. 1 (a) Determine the constant c. (b) Determine the expectation of X. (c) Explain how using the inverse-transform method we can simulate an outcome X. (d) Simulate 10000 via the inverse-transform method and show that this average is close to your answer in (b). 7. Let Y ∼ N(2, 5) and Z = 3Y − 4. Give the distribution of Z. Explain. Do the same for Y ∼ U(2, 5). 8. A certain multiple choice exams has 30 questions, each providing 3 choices. To pass the exam one needs at least 20 out of 30 correct answers. (a) Suppose a student knows the answers to 27 questions for certain, and fills in the remaining three questions “at random”. What is the probability that the student will get full marks? (b) Another student knows only 10 questions for certain and, like the student in (a), fills in the remaining questions at random. Calculate the probability that this student will pass. 9. (#) Implement a computer program that simulates the Snakes & Ladders game in Figure 1. Figure 1: A toy snakes & ladder game. Estimate how long (how many throws) the game lasts on average. Estimate the probability mass function for the length of the game. 10. (*) Consider a game in which a fair coin is tossed indefinitely. Every time Heads appears you move one metre to the right, but if Tails appears you move one metre to the left. You start at position 0. Let Yn be your position after n tosses, n = 1, 2, . . .. (a) Determine the pmf of Yn. (b) Determine the expectation of Yn. (c) Determine the variance of Yn. 2