Help....plz.

4:42 P 5 P - 29 ... OB/s U Mil mil @ 21% 9E-12 You have a thoroughly shuffled deck of 52 cards. Each time you choose one card from the deck. The drawn card is put back in the deck and all 52 cards are again thoroughly shuffled. You continue this procedure until you have seen all four different aces. What are the expected value and the standard deviation of the number of times you have to draw a card until you have seen all four different aces? 9E-13 A group of m people simultaneously enter an elevator at the ground floor. Each person randomly chooses one of the r floors 1, 2, ..., " as the exit floor, where the choices of the persons are independent of each other. The elevator only stops on a floor if at least one person wants to exit on that floor. No other people enter the elevator at any of the floors 1, 2, ..., r. What are the expected value and the standard deviation of the number of stops the elevator will make? 9E-14 (a) An integer is repeatedly drawn at random from 1, 2, . .., 10. What are the expected value and the standard deviation of the number of integers from 1, 2, ..., 10 that do not show up in 20 drawings? (b) In each drawing of the Lotto 6/45 six different integers are randomly chosen from 1, 2, ..., 45. What are the expected value and the standard deviation of the number of integers from 1, 2, ..., 45 that do not show up in 15 drawings? 9E-15 Take a random permutation of the integers 1, 2, ..., n. Let us say that the integers i and j with i / j are switched if the integer i occupies the jth position in the random permutation and the integer j the ith position. What is the expected value of the total number of switches? 9E-16 Twelve married couples participate in a tournament. The group of 24 people is randomly split into eight teams of three people each, where all possible splits are equally likely. What is the expected value of the number of teams with a married couple? 9E-17 Suppose you know that the hands of you and your bridge partner contain eight of the 13 spades in the deck. What is the probability of 3-2 split of the remaining five spades in the bridge hands of your opponents? 9E-18 You choose at random an integer from 1,2, ...,6. Next you roll a fair die until you get an outcome that is larger than or equal to the randomly chosen integer. What is the probability mass function of the number of times you will roll the die? What are the expected value and the standard deviation of the number of times you will roll the die? 9E-19 You have two coins. One coin is fair and the other is biased with probability p of heads. The first toss is done with the fair coin. At the 12 subsequent tosses the fair coin is used if the previous toss resulted in heads and the biased coin is used otherwise. What is the expected value of the number of heads in r tosses for r = 1, 2, ...? 9E-20 A bag contains R red balls and W white balls. Each time you take one ball out of the bag at random and without replacement. You stop as soon as all red balls have been taken out of the bag. What is the expected number of white balls remaining in the bag when you stop? 9E-21 Let the random variable X be defined by X = YZ, where Y and Z are independent random variables each taking on the values -1 and 1 with probabilities 0.5. Verify that X is independent of both Y and Z, but not of Y +Z. 9E-22 Let X and Y' be independent random variables, where X is binomially distributed with parameters n and p and Y is binomially distributed with parameters m and p. (a) Explain in terms of Bernoulli experiments that X + Y is binomially distributed with parameters n + m and p. Next give a formal proof. (b) Verify that for fixed k the probabilities P(X = j | X + Y = k) for j =0. .. ., k constitute a hypergeometric distribution. 9E-13 / radioactive source emits particles toward a Geiger counts The number of particles that are emitted in a given time interval is Poisson distributed with expected value A. An emitted particle is recorded by the75.9 76.5 77.0 88.9 (a) Find the P-value for the first alternative (p1 > p2) if a = 6. (a) Calculate range, IQR, and median of these (b) Find the P-value for the third alternative data. (Pi # p2) ifr = -0.2500. (b) Given that the mean of these data is (c) Determine both the P-value and r that sat- 63.50 (exactly) and the standard devia- isfy the following condition: The data are tion is 12.33, what proportion of the data statistically significant but not highly sta- lie within one standard deviation of the tistically significant for the second alter- mean? native (p1 p2) if a = 6. 2 (b) Find the P-value for the third alternative 7. Consider an unbalanced study with six subjects, (Pi # pz)ifr = -0.1761. identified as A, B, C, D, E and G. In the actual (c) Determine both the P-value and r that sat- study, isfy the following condition: The data are . Subjects A and B are assigned to the first statistically significant but not highly sta- reatment, and the other subjects are as- tistically significant for the second alter- signed to the second treatment. native (p1