E1X 2

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x b a

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0.5x2 b ` b a a

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b2 2

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The variance of X is ax a b

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a 2 a

V1X 2

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2 31b a2

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These results are summarized as follows.

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If X is a continuous uniform random variable over a E1X 2 1a b2 2 and

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b, 1b a2 2 12 (4-7)

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V1X 2

CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

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f(x)

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1 b a

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f(x) 0.05 a b x 0 5 10 15 20 x

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Figure 4-8 Continuous uniform probability density function.

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Figure 4-9 Probability for Example 4-9.

EXAMPLE 4-9

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Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [0, 20 mA], and assume that the probability density function of X is f 1x2 0.05, 0 x 20. What is the probability that a measurement of current is between 5 and 10 milliamperes The requested probability is shown as the shaded area in Fig. 4-9.

f 1x2 dx 510.052 0.25 0 and b 20. Therefore,

The mean and variance formulas can be applied with a E1X 2 10 mA and V1X 2

202 12

33.33 mA2

Consequently, the standard deviation of X is 5.77 mA. The cumulative distribution function of a continuous uniform random variable is obtained by integration. If a x b,

F1x2

1 1b

a2 du

x 1b

a 1b

Therefore, the complete description of the cumulative distribution function of a continuous uniform random variable is 0 1x 1 a2 1b x x x a b

F1x2

An example of F(x) for a continuous uniform random variable is shown in Fig. 4-6. EXERCISES FOR SECTION 4-5

4-31. Suppose X has a continuous uniform distribution over the interval [1.5, 5.5]. (a) Determine the mean, variance, and standard deviation of X. (b) What is P1X 2.52 4-32. Suppose X has a continuous uniform distribution over the interval 3 1, 1 4. (a) Determine the mean, variance, and standard deviation of X. (b) Determine the value for x such that P( x X x) 0.90. 4-33. The net weight in pounds of a packaged chemical herbicide is uniform for 49.75 x 50.25 pounds. (a) Determine the mean and variance of the weight of packages.

4-6 NORMAL DISTRIBUTION

(b) Determine the cumulative distribution function of the weight of packages. (c) Determine P1X 50.12. 4-34. The thickness of a ange on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters. (a) Determine the cumulative distribution function of ange thickness. (b) Determine the proportion of anges that exceeds 1.02 millimeters. (c) What thickness is exceeded by 90% of the anges (d) Determine the mean and variance of ange thickness. 4-35. Suppose the time it takes a data collection operator to ll out an electronic form for a database is uniformly between 1.5 and 2.2 minutes. (a) What is the mean and variance of the time it takes an operator to ll out the form (b) What is the probability that it will take less than two minutes to ll out the form (c) Determine the cumulative distribution function of the time it takes to ll out the form. 4-36. The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f 1x2 0.2 for 50 x 75 seconds.

(a) What percentage of tests require more than 70 seconds to complete. (b) What percentage of tests require less than one minute to complete. (c) Determine the mean and variance of the time to complete a test on a sample. 4-37. The thickness of photoresist applied to wafers in semiconductor manufacturing at a particular location on the wafer is uniformly distributed between 0.2050 and 0.2150 micrometers. (a) Determine the cumulative distribution function of photoresist thickness. (b) Determine the proportion of wafers that exceeds 0.2125 micrometers in photoresist thickness. (c) What thickness is exceeded by 10% of the wafers (d) Determine the mean and variance of photoresist thickness. 4-38. The probability density function of the time required to complete an assembly operation is f 1x2 0.1 for 30 x 40 seconds. (a) Determine the proportion of assemblies that requires more than 35 seconds to complete. (b) What time is exceeded by 90% of the assemblies (c) Determine the mean and variance of time of assembly.