1. In the Statistics for CS class of size 20, by the time of the final exam, 80% of the students mastered discrete probabilities, 70% mastered continuous probabilities, and 60% mastered both. For a student to pass the class at least a C, the student must be able to use at least one type of probabilities. How many students passed the class? Are two parts statistically independent?
2. Another year, out of 25 students, 80% of the students mastered discrete probabilities, 70% mastered continuous probabilities, and the corresponding events turned out to be independent. How many students passed the class?
3. An average student is well-prepared (A-level) for five out of seven topics. If an instructor gives a surprise quiz covering two of the topics, what percentage of students will get an A?
4. Suppose that for each class, 30% of the student are well-prepared (A-level), and different students are independent. What is the probability that in a class of 20, 19 will be well-prepared? What is the probability that the instructor will grade at least 10 papers until he finds a one which does not deserve an A?
5. The quiz has three yes-no questions, testing thee (randomly selected) topics out of five that the students studied. A student knows four out of these five topics; for a topic that a student does not know, he gives yes or no answer with equal probability. What are the expected number of mistakes on the quiz? What is the variance of this number?
6. Two students did not prepare well for a quiz. To answer a multiple-choice question with 4 possible answers, each student independently chooses a random number from 1 to 4, with equal probability. Let s be the sum of these numbers, and let p be their product. What is the probability that s = 5 if p = 6? if p = 5? Are the variables s and p independent? Explain your answer.
7. A professor (not me) lets a student take the test again and again until the student passes it. A student relies on luck and does not study between the tests, so his probability of succeeding is 50% on each test. On average, how many times will a student take the test until he passes?
8. On average, a student makes 3 mistakes on a test, with a standard deviation of 2. In a class of 50 students, what is a probability that overall, they will make less than 120 mistakes? Use Central Limit Theorem.
9. A student knows 5 topics out of 10. What is the probability that this student will get an A on a quiz which covers 4 out of 10 topics? 5 out of 10? 6 out of 10?
10. A random variable is distributed with the probability density f(x) = 1 + kx for x from the interval [0,1] and f(x) = 0 for all other x. Find: (a) the value k, (b) the cumulative distribution function for x, (c) the expected value of x, and (d) the probability that x does not exceed 0.5.
11. A computer is connected to two printers. If at least one of them works, we can still print. The time-to-repair of each printer is exponentially distributed with the mean value of 2 years. What is the probability that after 3 years, we will still be able to print?
12. For a printer-producing company, the probability that a printer needs repairs is 1%. In a sample of 1,000 printers, what is the expected value of number of faulty printers, and what is the standard deviation? What is the probability that no more than 20 printers are faulty (just give a formula, it is not necessary to get a numerical answer).
13. Two independent tasks are coming to the server. Their arrival times are uniformly distributed between 6 am and noon. Compute the expected time of the task which arrives first, and the expected time of the task which arrives second.
14. Derive a formula and explain how to generate a random variable with the density f(x) = (2/9) x on the interval [0,3] if you have a standard random number generator which produces values uniformly distributed on the interval [0,1]. Use the inverse transform method.
15. Suppose that we have a distribution in which we have 1 with probability θ and 0 with probability 1 − θ. Use method of moments and maximum likelihood to estimate the parameter θ from the following sample: 0, 1, 0, 1, 0, 0, 1, 1.
16. In a shipment of 100 computers, 3 turns out to be faulty. Construct a 90% confidence interval for the proportion of faulty printers.