Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Chapter 4 Continuous Random Variables and Probability Distributions Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Continuous Random Variables 2 • A continuous random variable is one which takes values in an uncountable set. • They are used to measure physical characteristics such as height, weight, time, volume, position, etc... Examples 1. Let Y be the height of a person (a real number). 2. Let X be the volume of juice in a can. 3. Let Y be the waiting time until the next person arrives at the server. Sec 4-1 Continuos Radom VariablesCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Probability Density Function Sec 4-2 Probability Distributions & Probability Density Functions 3 Note: For continuous PDF, it does not make sense to talk about P(X = x)!Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Example 4-1: Electric Current Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes(mA). Assume that the range of X is 4.9 ≤ x ≤ 5.1 and f(x) = 5. What is the probability that a current is less than 5mA? Answer: Sec 4-2 Probability Distributions & Probability Density Functions 4 Figure 4-4 P(X < 5) illustrated. 554.9 4.95 ( ) 5 0.5P X f x dx dx 5.14.954.95 5.1 ( ) 0.75P X f x dx Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Cumulative Distribution Functions Sec 4-3 Cumulative Distribution Functions 5 The cumulative distribution function is defined for all real numbers.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Example 4-3: Electric Current For the copper wire current measurement in Exercise 4-1, the cumulative distribution function consists of three expressions. Sec 4-3 Cumulative Distribution Functions 6 0x < 4.9 F (x ) = 5x - 24.54.9 ≤ x ≤ 5.11 5.1 ≤ xFigure 4-6 Cumulative distribution function The plot of F(x) is shown in Figure 4-6.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Probability Density Function from the Cumulative Distribution Function • The probability density function (PDF) is the derivative of the cumulative distribution function (CDF). • The cumulative distribution function (CDF) is the integral of the probability density function (PDF). Sec 4-3 Cumulative Distribution Functions 7 Given , as long as the derivative exists.dF xF x f xdxCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Exercise 4-5: Reaction Time • The time until a chemical reaction is complete (in milliseconds, ms) is approximated by this cumulative distribution function: • What is the Probability density function? • What proportion of reactions is complete within 200 ms? Sec 4-3 Cumulative Distribution Functions 8 0.010 for 01 for 0 xxxFxe 0.01 0.010 0 for 01 0.01 for 0 xxxdF xdxfxeedx dx 2200 200 1 0.8647P X F e Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Mean & Variance Sec 4-4 Mean & Variance of a Continuous Random Variable 9Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Example 4-6: Electric Current For the copper wire current measurement, the PDF is f(x) = 0.05 for 0 ≤ x ≤ 20. Find the mean and variance. Sec 4-4 Mean & Variance of a Continuous Random Variable 10 2020200203202000.051020.05 1010 33.333xE X x f x dxxV X x f x dx Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Mean of a Function of a Continuous Random Variable Sec 4-4 Mean & Variance of a Continuous Random Variable 11 If is a continuous random variable with a probability density function , (4-5)X f xE h x h x f x dxExample 4-7: Let X be the current measured in mA. The PDF is f(x) = 0.05 for 0 ≤ x ≤ 20. What is the expected value of power when the resistance is 100 ohms? Use the result that power in watts P = 10−6RI2, where I is the current in milliamperes and R is the resistance in ohms. Now, h(X) = 10−6100X2. Note the typo – forgot to multiply by 0.05! 20203420010 0.0001 0.2667 watts3xE h x x dx Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Continuous Uniform Distribution • This is the simplest continuous distribution and analogous to its discrete counterpart. • A continuous random variable X with probability density function f(x) = 1 / (b-a) for a ≤ x ≤ b Sec 4-5 Continuous Uniform Distribution 12 Figure 4-8 Continuous uniform Probability Density FunctionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Mean & Variance • Mean & variance are: Sec 4-5 Continuous Uniform Distribution 13 22 2and 12abEXbaVXCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Example 4-9: Uniform Current The random variable X has a continuous uniform distribution on [4.9, 5.1]. The probability density function of X is f(x) = 5, 4.9 ≤ x ≤ 5.1. What is the probability that a measurement of current is between 4.95 & 5.0 mA? Sec 4-5 Continuous Uniform Distribution 14 Figure 4-9 The mean and variance formulas can be applied with a = 4.9 and b = 5.1. Therefore, 220.25 mA and =0.0033 mA 12E X V X Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Cumulative distribution function of Uniform distribution Sec 4-5 Continuous Uniform Distribution 15 1The Cumulative distribution function is 0 1 xaxaF x dub a b axaF x x a b a a x bbx Figure 4-6 Cumulative distribution functionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved. Normal Distribution Sec 4-6 Normal Distribution 16Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. Empirical Rule For any normal random variable, P(μ – σ < X < μ + σ) = 0.6827 P(μ – 2σ < X < μ + 2σ) = 0.9545 P(μ – 3σ < X < μ + 3σ) = 0.9973
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