Название: Probability and Statistical Inference
Автор: Robert Bartoszynski
Издательство: John Wiley & Sons Limited
Жанр: Математика
isbn: 9781119243823
isbn:
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.
Example 1.9 Randomized Response
Let
The direct question reduced to something like “Are you a
Figure 1.4 Scheme of a randomized response.
The interviewer knows the answer “yes” or “no” but does not know whether it is the answer to the question about
One could wonder what is a possible advantage, if any, of not knowing the question asked and observing only the answer. This does not make sense if we need to know the truth about each individual respondent. However, one should remember that we are only after the overall frequency of
We are in fact “contaminating” the question by making the respondent answer either a
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