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Continuous Probability Distributions Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Probability for a Continuous Random Variable Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.1 Properties of a Normal Distribution • Continuous Random Variable • Symmetrical in shape (Bell shaped) • The probability of any given range of numbers is represented by the area under the curve for that range. • Probabilities for all normal distributions are determined using the Standard Normal Distribution. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Probability Density Function for Normal Distribution x 1 1 ( ) 2 f (x) e 2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing 2 Figure 6.2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.3 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.4 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.5 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.6 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Determining the Probability for a Standard Normal Random Variable • Figures 6.10-6.13 • P(- Z 1.62) = .5 + .4474 = .9474 • P(Z > 1.62) = 1 - P(- Z 1.62) = 1 - .9474 = .0526 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.10 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.11 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Determining the probability of any Normal Random Variable Fig 6.20 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Interpreting Z • Example 6.2 Z = - 0.8 means that the value 360 is .8 standard deviations below the mean. • A positive value of Z designates how may standard deviations () X is to the right of the mean (). • A negative value of Z designates how may standard deviations () X is to the left of the mean (). Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Example 6.5 Referring to Example 6.2, after how many hours will 80% of the Evergol bulbs burn out? P(Z < .84) = .5 + .2995 = .7995 .8 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.26 Figure 6.26 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing x o 400 Z .84 50 x o 400 50(.84) 42 x o 400 42 442 Continuous Uniform Distribution • The probability of a given range of values is proportional to the width of the range. • Distribution Mean: ab 2 • Standard Deviation: Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing b– a 12 Figure 6.35 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Figure 6.36 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Exponential Distribution Applications: • Time between arrivals to a queue (e.g. time between people arriving at a line to check out in a department store. (People, machines, or telephone calls may wait in a queue) • Lifetime of components in a machine Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Mean and Standard Deviation P(X x0 ) 1 – e–Ax 0 where A 1/ , Mean: Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing and 1 = A Standard Deviation: Introduction to Business Statistics, 5e for x0 0 1 . A Figure 6.39 P (X x0 ) 1 – e– Ax0 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing for x0 0 1 1 where A 1 / , = , and . A A