Probability Distributions - Oxford University Press

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178 Business Statistics Using Excel Expected value The expected value of a random data variable indicates its population average value. Normal distribution The normal distribution is a symmetrical, bell-shaped curve, centred at its expected value. X » Understand the concept of a probability distribution. » Calculate using this distribution the expected value and a measure of spread. » Have an introduction to discrete distributions. » Understand when to apply the binomial distribution. » Solve simple problems using both tree diagrams and the binomial formula. » Understand when to apply the Poisson distribution. » Solve simple problems using the Poisson formula. » Have an introduction to continuous distributions. » Use the normal distribution to calculate the values of a variable that correspond to a particular probability. » Calculate as one parameter of the normal distribution if the other parameters are known. » Use the normal distribution to calculate the probability that a variable has a value between specific limits. » Understand when to apply approximations to simplify the solution process. » Have an introduction to the uniform distribution. » Have an introduction to the Student’s t distribution. » Have an introduction to the chi square distribution. » Have an introduction to the F distribution. » Solve problems using Microsoft Excel. 5.1 Introduction to Probability 5.1.1 Basic ideas There are a number of words and phrases that encapsulate the basic concept of probability: chance, probable, odds and so on. In all cases we are faced with a degree of uncertainty and concerned with the likelihood of a particular event happening. Statistically these words and phrases are too vague; we need some measure of likelihood of an event occurring. This measure is termed probability and is measured on a scale ranging between 0 and 1. From Figure 5.1 we ob serve that the probabil Not possible Figure 5.1 Equal possible outcome Certainty 0 0.5 Probability value, P 1 ity values lie between 0 and 1 with 0 representing no possibility of the event occurring and 1 representing the probability is certain to occur. In reality the value of the probability will lie between 0 and 1. In order to determine a probability of an event occurring, data has to be obtained. This can be achieved through, for example, experience or
observation or empirical methods. The procedure or situation that produces a def nite result (or outcome) is termed a random experiment. For example tossing a coin, rolling a die, recording the income of a factory worker, determining defective items on an assembly line are all examples of ‘experiments’. The characteristics of random experiments are: • • • Each experiment is repeatable. All possible outcomes can be described. Although individual outcomes appear haphazard, continual repeats of the experiment will produce a regular pattern. The result of an experiment is called an outcome. It is the single possible result of an experiment, e.g. tossing a coin produces ‘a head’, rolling a die gives a ‘3’. If we accept the proposition that an experiment can produce a fnite number of outcomes then we could in theory defne all these outcomes. The set of all possible outcomes is defned as the sample space. For example the experiment of rolling a die could produce the outcomes: 1, 2, 3, 4, 5, and 6; which would thus defne the sample space. Another basic notion is the concept of an event and is simply a set of possible outcomes i.e. an event is a subset of the sample space. For example take the experiment of rolling a die, the event of obtaining an even number would be defned as the subset {2, 4, 6}. Furthermore, two events are said to be mutually exclusive if they cannot occur together. Thus in rolling a die, the event ‘obtaining a two’ is mutually exclusive of the event ‘obtaining a three’. The event ‘obtaining a two’ and the event ‘obtaining an even number’ are not mutually exclusive since both can occur together i.e. {2} is a subset of {2, 4, 6}. Student Exercises X5.1 Give an appropriate sample space for each of the following experiments: (a) A card is chosen at random from a pack of cards. (b) A person is chosen at random from a group containing 5 females and 6 males. (c) A football team records the results of each of two games as ‘win’, ‘draw’, or ‘lose’. X5.2 A dart is thrown at a board and is likely to land on any one of eight squares numbered 1 to 8 inclusive. A represents the event the dart lands in square 5 or 8. B represents the event the dart lands in square 2, 3, or 4. C represents the event the dart lands in square 1, 2, 5, or 6. Which two events are mutually exclusive? 5.1.2 Relative frequency Suppose we perform the experiment of throwing a die and note the score obtained. We repeat the experiment a large number of times, say 1000, and note the number of times each score was obtained. For each number we could derive the ratio of occurrence that an event A will happen (m) to the total number of experiments (n = 1000). This ratio is called Probability Distributions 179
Page 1: Probability Distributions 5 » Over
Page 5 and 6: Determine the following probabiliti
Page 7 and 8: Example 5.3 Suppose a fair die is t
Page 9 and 10: Figure 5.5 ➜ Excel Solution Milea
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Page 13 and 14: outes that we can achieve 3, 2, 1,
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Page 19 and 20: them. This is possible by creating
Page 21 and 22: ❉ Interpretation We observe that
Page 23 and 24: Example 5.14 A local authority inst
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Page 27 and 28: Note 1. This problem corresponds to
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Page 31 and 32: (c) Figure 5.32 illustrates a right
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Page 41 and 42: X5.15 Attendance at a cinema has be
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Normal Lower X 1 = Cell D13 Formula
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Note (a) Binomial solution P(X = 35
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Normal X = Cell D16 Value P(X ≥ 7
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having problems with the production
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SD( X) = VAR( X ) (5.8) 1 fX 2 exp
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Probability Distributions - Oxford University Press

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