Probability and Statistical Inference. Robert Bartoszynski
Чтение книги онлайн.

Читать онлайн книгу Probability and Statistical Inference - Robert Bartoszynski страница 15

Название: Probability and Statistical Inference

Автор: Robert Bartoszynski

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119243823

isbn:

СКАЧАТЬ alt="images"/> exceeds images” would lead to maximizing the (unobservable) number of persons accepted whose true level of skill images exceeds images. Naturally, to find such a solution, one needs to understand statistical relation between observable images and unobservable images.

      Another example when the points in the sample space are only partially observable concerns studies of incidence of activities about which one may hesitate to respond truthfully, or even to respond at all. These are typically studies related to sexual habits or preferences, abortion, law and tax violation, drug use, and so on.

      Let images be the activity analyzed, and assume that the researcher is interested in the frequency of persons who ever participated in activity images (for simplicity, we will call them images‐persons). It ought to be stressed that the objective is not to identify the images‐persons, but only to find the proportion of such persons in the population.

Flowchart depicting the scheme of a randomized response, where the respondent is given a pair of distinguishable dice in two different shades, with one depicting an odd face and the other an even face.

      The interviewer knows the answer “yes” or “no” but does not know whether it is the answer to the question about images or the question about the white die. Here a natural sample space consists of points images where images and images are outcomes on green and white die, respectively, while images is 1 or 0 depending on whether or not the respondent is a images‐person. We have images= “yes” if images and images or 5 for any images, or if images and images for any images. In all other cases, images “no.”

      We are in fact “contaminating” the question by making the respondent answer either a images‐question or some other auxiliary question. But this is a “controlled contamination”: we know how often (on average) the respondents answer the auxiliary question, and how often their answer is “yes.” Consequently, as we will see in Chapter 11, we can still make an inference about the proportion of images‐persons based on the observed responses.

      Problems

      1 1.2.1 List all sample points in sample spaces for the following experiments: (i) We toss a balanced coin.1 If heads come up, we toss a die. Otherwise, we toss the coin two more times. (ii) A coin is tossed until the total of two tails occurs, but no more than four times (i.e., a coin is tossed until the second tail or fourth toss, whichever comes first).

      2 1.2.2 Alice, Bob, Carl, and Diana enter the elevator on the first floor of a four‐story building. Each of them leaves the elevator on either the second, third, or fourth floor. (i) Describe the sample space without listing all sample points. (ii) List all sample points such that Carl and Diana leave the elevator on the third floor. (iii) List all sample points if Carl and Diana leave the elevator at the same floor.

      3 1.2.3 An urn contains five chips, labeled . Three chips are selected. List all outcomes included in the event “the second largest number drawn was 3.”

      4 1.2.4 In a game of craps, the player rolls a pair of dice. If he gets a total of 7 or 11, he wins at once; if the total is 2, 3, or 12, he loses at once. Otherwise, the sum, say , is his “point,” and he keeps rolling dice until either he rolls another (in which case he wins) or he rolls a 7 in which case he loses. Describe the event “the player wins with a point of 5.”

      5 1.2.5 The experiment consists of placing six balls in three boxes. List all outcomes in the sample space if: (i) The balls are indistinguishable, but the boxes are distinguishable. (Hint: There are 28 different placements.) (ii) Neither the balls nor the boxes are distinguishable. (iii) Two balls are white and СКАЧАТЬ