Probability and Statistics Random Sampling of Questions

Random Sampling of Questions 1. The probability that rain is followed by rain is 0.8, a sunny day is followed by rain is 0.6. Find the probability that one has two rainy days then two sunny days. 2. Let: f (x) = k|x − 2|, for x = −1, 0, 1, 3 • Find k so that f is a PDF. • Find the expected value of X. • Find the expected value of X 2 . 3. A balanced die is tossed twice. Let A be the event that an even number comes up on the first toss. Let B be the event that an even number comes up on the second. Let C be the event that the first two tosses gave the same number. Are A, B, C • pairwise independent • independent? 4. State Bayes’ Theorem. 5. If A, B are independent prove that A0 and B 0 are independent. 6. What is the difference between saying that A, B, C are independent events, versus saying that X, Y, Z are independent random variables? 7. In a poker game, 5 cards are dealt from a deck of 52. What is the probability of getting a four-of-a-kind? 8. If the joint PDF is given by: f (x, y, z) = xyz for x = 1, 2, 3; y = 1, 2, 3; z = 1, 2 108 (a) Find the joint marginal distribution of X and Y . (b) Find the conditional distribution of Z given X = 1 and Y = 2. (c) Find the marginal distribution of X. (d) Are X, Y, Z independent random variables? Prove or Disprove. 9. How many ways can a college team playing 10 games end up with 5 wins, 4 losses and a tie? 10. A carton contains 15 light bulbs of which one is defective. In how many ways can you choose 3 of the bulbs and: (a) Get the one that is defective? (b) Not get the one that is defective? (c) Find the probability of choosing the defective bulb. 11. Given that 8% of the population has diabetes, a health department comes in and gives tests. It correctly diagnoses 95% of all persons with diabetes as having the disease, and incorrectly diagnoses 2% of all persons without diabetes as having the disease. (a) Find the probability that the health department will diagnose someone in the population as having the disease. 1 (b) Find the probability that someone diagnosed by the health department as having the disease actually has it. 1 for 0 < x < 3 (zero elsewhere). Find E(X) and E(X 2 ), x ln(3) then use the results to find E(2X 2 − X + 1) 12. Let f (x) = 13. Experiment: Roll three standard dice. Observe the outcomes. (a) How many ways can the three dice all come up with the same number of points? (b) How many ways can two of the three come up with the same number, but the third die is different? (c) How many ways can all three dice come up with different numbers? 2x for x = 1, 2, 3, . . . k can serve as the probabilk(k + 1) ity distribution of a random variable with the given range. 14. Verify that f (x) = 15. State the Law of the Unconscious Statistician. 16. We have two men and four women making up a committee of three. • What is the probability of choosing no men? one man? two men? • Can you come up with a probability distribution function (PDF) for this using binomials? 17. How many ways can a bakery distribute its 7 unsold apple pies to 4 food banks? How does this change if every food bank must get at least one? 18. If I have four skirts, seven blouses and three sweaters, how many ways can I pick 2 skirts, three blouses and one sweater to take along on a trip? 19. Two cards are randomly drawn from a deck of 52. Find the probability that both cards will be greater than 3 and less than 8. 20. If the joint density of X, Y is given by  4xy if 0 < x < 1, 0 < y < 1 f (x, y) = 0 elsewhere (a) Compute f (x|y) and f (y|x). (b) Set up an integral to determine P (X + Y < 1) (c) Find F (x, y). Be careful of the regions (like we did in class). (d) Are X, Y independent random variables? (e) Set up an integral to determine E(X) and E(Y ). Is the product the same as E(XY )? Do you think that always happens? 2