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Since the outcome of an experiment is not known in advance, it is important to determine the set of all possible outcomes. considerations of probability. Denition 1. The sample space, denoted by S, is the set of all outcomes of an experiment. The elements of the sample space are called elementary outcomes, or sample points. Example 2. In Example 1 the sample space S has 62 = 36 sample points in the case of two tosses, and 63 = 216 points in the case of three tosses of a die. This set, called the sample space, forms the conceptual framework for all further
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we let A and B denote the two persons, and then take as S the set of outcomes represented by 12 ideograms in Figure 1.1.
ECONOMETRICS (I)
Course one Experiments, Sample Spaces, and Events
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Introduction
Overviews of the probability and statistics? History. Applications: biological, medical, social, physical sciences ( engineering, humanities ( economics and nance. Why we learn this course? (1) To make better predictions and decisions under uncertainty. (2) To better learn econometric techniques in next term. How to study? (1) Reading, (2) homework, and (3) understanding gradually.
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EXAMPLE 1.1
Consider an experiment consisting of two tosses of a regular die. An outcome isExample 1. Consider an by a pair of numbers that of two tosses upper most naturally represented experiment consisting turn up on the of a faces of the die so ). An outcomeais most naturally represented .by ,a pair that they form pair (2, with 5 ,y = 1 , 2 , . . 6 (see y), regular die ( Table 1.1). that turn up on the upper faces of the die so that they form of numbers
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Example 5. Let the experiment consist of recording the lifetime of a piece of equipment, say a light bulb ( ). An outcome here is the time until the bulb bums out. An outcome typically will be represented by a number t ≥ 0 (t = 0 if the bulb is not working at the start), and therefore S is the nonnegative part of the real axis. In practice, t is measured with some precision (in hours, days, etc.), so one might instead take S = {0, 1, 2, . . .}. Which of these choices is better depends on the type of subsequent analysis. Example 6. Two persons enter a cafeteria and sit at a square table, with one chair on each of its sides. Suppose we are interested in the event “they sit at a corner” (as opposed to sitting across from one another). To construct the sample space, we let A and B denote the two persons, and then take as S the set of outcomes represented by 12 ideograms in Figure 1.
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Example 3. Suppose that the only available information about the numbers, those that turn up on the upper faces of the die, is their sum. In such a case as outcomes we take 11 possible values of the sum so that S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. For instance, all outcomes on the diagonal of Table 1–(6, 1), (5, 2), (4, 3), (3, 4), (2, 5), and (1, 6)-are represented by the same value 7. Example 4. If we are interested in the number of accidents that occur at a given intersection within a month, the sample space might be taken as the set S = {0, 1, 2, . . .} consisting of all nonnegative integers. Realistically, there is a practical limit, say 1000, of the monthly numbers of accidents at this particular intersection. Although one may think that it is simpler to take the sample space S = {0, 1, 2, ..., 1000}, it turns out that it is often much simpler to take the innite sample space if the “practical bound” is not very precise.
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Examples of a random experiment The tossing of a coin. outcomes, heads or tails. The roll of a die. The experiment can yield six possible outcomes, this outcome is the number 1 to 6 as the die faces are labelled. The selection of a numbered ball (1-50) in an urn. The experiment can yield 50 possible outcomes. The time difference between two messages arriving at a message centre. This experiment can yield any number of possible outcomes. outcomes. The experiment can yield two possible
a pair (x, y), with x, y 1, 2, . , 6. Table 1.1 Outcomes on = Pair .of.Dice a
Table 1 Outcomes on a Pair of Dice
In the case of an experiment of tossing a die three times, the outcomes will be tripletsone 2 ,y, Experiments, 2 , y, Spaces, and Eventsintegers between 1 and 6. z ) , with Sample and z being c WISE Course ( 5
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1.1.
Random experimentSample Space
A random experiment is an experiment, trial, or observation that can be repeated numerous times under the same conditions. The outcome of an individual random experiment must be independent and identically distributed. It must in no way be affected by any previous outcome and cannot be predicted with certainty. ; ; ;