chap 7 common continuous probability distributions · chap 1 intro to computers and the fortran...
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Chap 7 Common continuous probability distributions
Applied Statistics by John Neter, William Wasserman, and G. A. Whitmore
Prepared by Walter Chen, Dept. of Civil Engineering, NTUTFor classroom teaching purpose
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Examples of normal distributions The temperature X at noon on August 15 in a
southeastern city The weight X of a metal ingot produced in a
smelter The height X of women who are 20-29 years old
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Density function
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Characteristics of normal distribution Two parameters
µ (mu) σ (sigma)
Bell shaped Symmetrical
Centered at µ σ determines the spread of the distribution
Almost all of the probability in a normal distribution is located in a limited range about its mean
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Mean and varianceNotation: N(μ, σ2)For example, N(100, 20)
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Standard normal probability distribution Is a particular member of the family of normal
distributions Mean = 0 Standard deviation = 1 Standard normal variable Z
Z = N(0, 1)
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Theorem Any linear function of a normal random variable is
also a normal random variable Therefore, any normal random variable can be
transformed into the standard normal variable The probability table for the standard normal
distribution can be used for all normal distributions
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X → Z
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Standard normal probability table and examples
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Percentiles
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Probabilities for any normal distribution
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Descriptions for Figure 7.5 and 7.6
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Probability limits
The Six Sigma Way: How GE, Motorola, and Other Top Companies are Honing Their Performance
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Cumulative probability function for a normal distribution
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Sum of independent normal RVs
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Example
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Central limit theorem Under very general conditions, the distribution of
the sum of a number of random variables converges to, or approaches, the normal distribution as the number of variables in the sum becomes large
The theorem does not require that the random variables entering the sum have the same distribution function or even that they be entirely independent