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p69_exercises-3_1

# p69_exercises-3_1 - f1 RCISES FOR SECTION 3—1 each of the...

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Unformatted text preview: f1 RCISES FOR SECTION 3—1 each of the following exercises, determine the range (pos- ues) of the random variable. 3 . The random variable is the number of nonconforming sol- connections on a printed circuit board with 1000 connections. ,_ In a voice communication system with 50 lines, the ran— “‘5: variable is the number of lines in use at a particular time. . An electronic scale that displays weights to the nearest .1011 is used to weigh packages. The display shows only ﬁve . Any weight greater than the display can indicate is i“ 1 as 99999. The random variable is the displayed weight. A batch of 500 machined parts contains 10 that do not :1: urm to customer requirements. The random variable is the her of parts in a sample of 5 parts that do not conform to m» er requirements. it A batch of 500 machined parts contains 10 that do not {1 rm to customer requirements. Parts are selected succes— , without replacement, until a nonconforming part is ob- The random variable is the number of parts selected. The random variable is the moisture content of a lot of material, measured to the nearest percentage point. MASS FUNCTIONS 3—2 PROBABILITY DISTRIBUTIONS AND PROBABILITY MASS FUNCTIONS 69 3—7. The random variable is the number of surface ﬂaws in a large coil of galvanized steel. 3—8. The random variable is the number of computer clock cycles required to complete a selected arithmetic calculation. 3—9. An order for an automobile can select the base model or add any number of 15 options. The random variable is the number of options selected in an order. 3—10. Wood paneling can be ordered in thicknesses of 1/ 8, l / 4, or 3 / 8 inch. The random variable is the total thickness of paneling in two orders. 3—1 1. A group of 10,000 people are tested for a gene called Iﬁ202 that has been found to increase the risk for lupus. The random variable is the number of people who carry the gene. 3—12. In an acid-base titration, the milliliters of base that are needed to reach equivalence are measured to the nearest milli- liter between 0.1 and 0.15 liters (inclusive). 3—13. The number of mutations in a nucleotide sequence of length 40,000 in a DNA strand after exposure to radiation is measured. Each nucleotide may be mutated. PROBABILITY DISTRIBUTIONS AND PROBABILITY '— Random variables are so important in random experiments that sometimes we essentially ‘ ignore the original sample space of the experiment and focus on the probability distribution of the random variable. For example, in Example 3-1, our analysis might focus exclusively on the integers {0, 1, . . . , 48} in the range of X. In Example 3-2, we might summarize the ran- ? * 'LE 34 ('gital Channel P(X = 0) = 0.6561 P(X= 3) = 0.0036 dom experiment in terms of the three possible values of X, namely {0, 1, 2}. In this manner, a random variable can simplify the description and analysis of a random experiment. The probability distribution of a random variable X is a description of the probabilities associated with the possible values of X. For a discrete random variable, the distribution is often speciﬁed by just a list of the possible values along with the probability of each. In some cases, it is convenient to express the probability in terms of a formula. There is a chance that a bit transmitted through a digital transmission channel is received in error. LetX equal the number of bits in error in the next four bits transmitted. The possible values forX are {0, l, 2, 3, 4}. Based on a model for the errors that is presented in the following section, probabilities for these values will be determined. Suppose that the probabilities are P(X= 1) = 0.2916 P(X = 4) = 0.0001 P(X = 2) = 0.0486 The probability distribution of X is speciﬁed by the possible values along with the probability of each. A graphical description of the probability distribution of X is shown in Fig. 3-1. Suppose a loading on a long, thin beam places mass only at discrete points. See Fig. 3-2. The loading can be described by a function that species the mass at each of the discrete ...
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