RANDOM VARIABLES

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We often summarize the outcome from a random experiment by a simple number. In many of the examples of random experiments that we have considered, the sample space has been a description of possible outcomes. In some cases, descriptions of outcomes are suf cient, but in other cases, it is useful to associate a number with each outcome in the sample space. Because the particular outcome of the experiment is not known in advance, the resulting value of our variable is not known in advance. For this reason, the variable that associates a number with the outcome of a random experiment is referred to as a random variable.

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CHAPTER 2 PROBABILITY

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De nition A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment. A random variable is denoted by an uppercase letter such as X. After an experiment is conducted, the measured value of the random variable is denoted by a lowercase letter such as x 70 milliamperes.

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Sometimes a measurement (such as current in a copper wire or length of a machined part) can assume any value in an interval of real numbers (at least theoretically). Then arbitrary precision in the measurement is possible. Of course, in practice, we might round off to the nearest tenth or hundredth of a unit. The random variable that represents this measurement is said to be a continuous random variable. The range of the random variable includes all values in an interval of real numbers; that is, the range can be thought of as a continuum. In other experiments, we might record a count such as the number of transmitted bits that are received in error. Then the measurement is limited to integers. Or we might record that a proportion such as 0.0042 of the 10,000 transmitted bits were received in error. Then the measurement is fractional, but it is still limited to discrete points on the real line. Whenever the measurement is limited to discrete points on the real line, the random variable is said to be a discrete random variable. De nition A discrete random variable is a random variable with a nite (or countably in nite) range. A continuous random variable is a random variable with an interval (either nite or in nite) of real numbers for its range.

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In some cases, the random variable X is actually discrete but, because the range of possible values is so large, it might be more convenient to analyze X as a continuous random variable. For example, suppose that current measurements are read from a digital instrument that displays the current to the nearest one-hundredth of a milliampere. Because the possible measurements are limited, the random variable is discrete. However, it might be a more convenient, simple approximation to assume that the current measurements are values of a continuous random variable.

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Examples of Random Variables

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Examples of continuous random variables: electrical current, length, pressure, temperature, time, voltage, weight Examples of discrete random variables: number of scratches on a surface, proportion of defective parts among 1000 tested, number of transmitted bits received in error.

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EXERCISES FOR SECTION 2-8

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2-100. Decide whether a discrete or continuous random variable is the best model for each of the following variables: (a) The time until a projectile returns to earth. (b) The number of times a transistor in a computer memory changes state in one operation. (c) The volume of gasoline that is lost to evaporation during the lling of a gas tank.

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2-8 RANDOM VARIABLES

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(d) The outside diameter of a machined shaft. (e) The number of cracks exceeding one-half inch in 10 miles of an interstate highway. (f) The weight of an injection-molded plastic part. (g) The number of molecules in a sample of gas. (h) The concentration of output from a reactor. (i) The current in an electronic circuit.

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