5_Continuous - 5 Some Well-Known Continuous Probability...

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5 Some Well-Known Continuous Probability Distributions Chapter Outline 5.1 Introduction 5.2 Normal and Standard Normal Distributions 5.3 Rectangular Distribution 5.4 Exponential Distribution 5.5 Gamma and Erlang Distributions 5.6 Weibull and Rayleigh Distributions 5.7 Lognormal Distribution 5.8 Beta Distribution 5.9 Chi-Squared Distribution 5.10 T Distribution 5.11 F Distribution 5.12 Summary
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2 Chapter 5 Some Well-Known Continuous Probability Distributions 5.1 Introduction Continuous Distributions General and Sampling Dist. General Dist. Normal and Standard Normal Rectangular, Exponential, Gamma, Erlang, Weibull Rayleigh,Lognormal,Beta Sampling Dist. Chi-Square, T, F Figure 5.1: Three type of continuous probability distribution 5.2 Normal and Standard Normal Distributions Defnition 5.1 If the pdf of a random variable X is f X (x) = 1 2 πσ e ( 1 / 2 ) [ (x μ)/σ ] 2 , −∞ <x< , where π = 3 . 1415 ..., e = 2 . 7182 ... , −∞ <μ< , σ 0, then X is called a normal distribution; X normal (μ, σ 2 ) ,or X N (μ, σ 2 ) Defnition 5.2 If the pdf of a random variable Z is f Z (z) = 1 2 π e ( 1 / 2 )z 2 , −∞ <z< , where π = 3 . 1415 ; e = 2 . 7182 ,then Z is called a standard normal distribution; Z normal ( 0 , 1 ) Z N ( 0 , 1 ) .
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5.2 Normal and Standard Normal Distributions 3 Parameters of Normal Distribution: μ and σ 2 Theorem 5.1 The mean and variance of a normal distribution: If X N (μ, σ 2 ) ,thenE (X) = μ, V (X) = σ 2 f(x) x μ 1 = μ 2 σ 2 2 σ 2 1 Figure 5.2: Two normal curves : μ 1 = μ 2 , σ 2 1 2 2 The curve with the larger standard deviation is lower and spreads out farther. x μ 1 μ 2 σ 2 1 σ 2 2 Figure 5.3: Two normal curves : μ 1 2 , σ 2 1 = σ 2 2 The curve with the larger mean is centered to the right.
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4 Chapter 5 Some Well-Known Continuous Probability Distributions Figures of Normal Distribution We list the following properties of the normal curve: (a) The normal curve is a bell-shape curve. (b) The curve is symmetric about a vertical axis through the mean μ . (c) The normal curve approaches the horizontal axis asymptoti- cally as we proceed in either direction away from the mean. (d)Thecurvehasitspointsofin±ectionat x = μ ± σ ,isconcave downward if μ σ<x<μ + σ , and is concave upward otherwise. (e) The value of the peak of a normal curve is f(x = μ) = 1 2 πσ . (f) The total area under the curve and above the horizontal axis is equal to 1. f(x) x in±ection in±ection μ σ μ μ + σ 1 2 Figure 5.4: N( μ , σ 2 ) The peak of a normal curve is located at x = μ , the value is 1 2 ;andits points of in±ection are located at x = μ σ and x = μ + σ . Example 5.1 Let N(0, σ = 0.5), N(0, σ = 1), N(0, σ =2)bedrawnonthe same graph. Also N(1,1), N(0,1), N(-1,1) are drawn together on another graph. From the graphs, ²nd out their mean, median, mode, variance, skewness, and kurtosis.
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5.2 Normal and Standard Normal Distributions 5 f(x) x infection infection - 101 0 . 4 Figure 5.5: N( μ = 0, σ 2 = 1) The peak o± a standard normal curve is located at x = 0, the value is 1 2 π ' 0 . 4; and itspointso± infection are located at x =− 1and x = 1.
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5_Continuous - 5 Some Well-Known Continuous Probability...

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