Probability and Statistical Inference. Robert Bartoszynski

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СКАЧАТЬ style="font-size:15px;">      6 1.2.6 John and Mary plan to meet each other. Each of them is to arrive at the meeting place at some time between 5 p.m. and 6 p.m. John is to wait 20 minutes (or until 6 p.m., whichever comes first), and then leave if Mary does not show up. Mary will wait only 5 minutes (or until 6 p.m., whichever comes first), and then leave if John does not show up. Letting and denote the arrival times of John and Mary, determine the sample space and describe events (i)–(viii) by drawing pictures, or by appropriate inequalities for and . If you think that the description is impossible, say so. (i) John arrives before Mary does. (ii) John and Mary meet. (iii) Either Mary comes first, or they do not meet. (iv) Mary comes first, but they do not meet. (v) John comes very late. (vi) They arrive less than 15 minutes apart, and they do not meet. (vii) Mary arrives at 5:15 p.m. and meets John, who is already there. (viii) They almost miss one another.Problems 1.2.7–1.2.8 show an importance of the sample space selection.

      7 1.2.7 Let be the experiment consisting of tossing a coin three times, with H and T standing for heads and tails, respectively.The following set of outcomes is an incomplete list of the points of the sample space of the experiment : {HHH, HTT, TTT, HHT, HTH, THH}. Use a tree diagram to find the missing outcomes.An alternative sample space for the same experiment consists of the following four outcomes: no heads , one head , two heads and three heads . Which of the following events can be described as subsets of but not as subsets of ? More than two heads. Head on the second toss. More tails than heads. At least one tail, with head on the last toss. At least two faces the same. Head and tail alternate.Still another sample space for the experiment consists of the four outcomes and . The first coordinate is 1 if the first two tosses show the same face and 0 otherwise; the second coordinate is 1 if the last two tosses show the same face, and 0 otherwise. For instance, if we observe HHT, the outcome is . List the outcomes of that belong to the event of .Which of the following events can be represented as subsets of , but cannot be represented as subsets of ? First and third toss show the same face. Heads on all tosses. All faces the same. Each face appears at least once. More heads than tails.

      8 1.2.8 Let be an experiment consisting of tossing a die twice. Let be the sample space with sample points with and being the numbers of dots that appear in the first and second toss, respectively.(i) Let be the sample space for the experiment consisting of all possible sums so that . Which of the following events can be defined as subsets of but not of ? One face odd, the other even. Both faces even. Faces different. Result on the first toss less than the result on the second. Product greater than 10. Product greater than 30.(ii) Let be the sample space for the experiment consisting of all possible absolute values of the difference so that . Which of the following events can be defined as subsets of but not of ? One face shows twice as many dots as the other, Faces the same, One face shows six times as many dots as the other, One face odd, the other even, The ratio of the numbers of dots on the faces is different from 1.

      9 1.2.9 Referring to Example 1.9, suppose that we modify it as follows: The respondent tosses a green die (with the outcome unknown to the interviewer). If the outcome is odd, he responds to the Q‐question; otherwise, he responds to the question “Were you born in April?” Again, the interviewer observes only the answer “yes” or “no.” Apart from the obvious difference in the frequency of the answer “yes” to the auxiliary question (on the average 1 in 12 instead of 1 in 6), are there any essential differences between this scheme and the scheme in Example 1.9? Explain your answer.

1.3 Algebra of Events

      Next, we introduce concepts that will allow us to form composite events out of simpler ones. We begin with the relations of inclusion and equality.

      Definition 1.3.1 The event is contained in the event , or contains , if every sample point of is also a sample point of . Whenever this is true, we will write , or equivalently, .

      An alternative terminology here is that implies (or entails) .

Definition 1.3.2 Two events and are said to be equal, , if and .

      It follows that two events are equal if they consist of exactly the same sample points.

      Example 1.10

      Consider two tosses of a coin, and the corresponding sample space consisting of four outcomes: HH, HT, TH, and TT. The event “heads in the first toss” HH, HT} is contained in the event “at least one head” HH, HT, TH}. The events “the results alternate” and “at least one head and one tail” imply one another, and hence are equal.

Definition 1.3.3 The set containing no elements is called the empty set and is denoted by . The event corresponding to is called a null (impossible) event.

Example 1.11 ^*2

      The reader may wonder whether it is correct to use the definite article in the definition above and speak of “the empty set,” since it would appear that there may be many different empty sets. For instance, the set of all kings of the United States and the set of all real numbers such that are both empty, but one consists of people and the other of numbers, so they cannot be equal. This is not so, however, as is shown by the following formal argument (to appreciate this argument, one needs some training СКАЧАТЬ