ioe265f11-Lec10

ioe265f11-Lec10 - Lec 10 Normal Distribution Topics I...

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Lec 10 - Normal Distribution IOE 265 F11 1 1 Normal Distribution 2 Topics I. Properties of Normal Distribution II. Probability distributions for Normal Probability density function Cumulative distribution function III. Z Values – Standardizing Variables IV. Normal Approximation to Other Distributions

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Lec 10 - Normal Distribution IOE 265 F11 2 3 I. Normal Distribution Properties The normal distribution is undoubtedly the most important and useful one of them all! Mean, median, mode all coincide. Symmetrical. Notation: X ~ N ( , 2 ) 4 Applications Normal Distributions are EVERYWHERE! Examples: IQ Scores Measurement Error Economic Measures Most outputs of a random process that only have inherent variation present (SPC Charts). Combinations of other distributions (Central Limit Theorem). In fact, problem solving in many areas is often trying to figure out why data are NOT normally distributed.
Lec 10 - Normal Distribution IOE 265 F11 3 5 What is so important about knowing the distribution? If data are normally distributed, then we can make predictions about events given a sample from a population! Example: Copy Machine Breakdowns X – # copies until machine breakdown and X~N( =8000, 2 =500 2 ) Given a distribution, we can make predictions / assertions about the likelihood of a copy machine breakdown for some: Interval ~ Pr(7500 < X < 8500) cdf 6 II. Probability Distributions Probability Density Function for Normal Example: Suppose X~N(8000,500 2 ), find f(X=8000) Suppose X~N(8000,1 2 ) , find f(X=8000) Notice the scaling effect ~ remember condition 2 of pdf x e x f x 2 1 ) , ; ( ) 2 /( ) ( 2 2 0 and

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Lec 10 - Normal Distribution IOE 265 F11 4 7 III. Standardizing a Variable To simplify the integration of f (x, , ), we will map our data into standardized form, Z-score. Z-score standardizes a variable, X, so that it has a mean = 0, and variance = 1. We may standardize our variable, X, by performing the following manipulation: Z = X - 8 Normal Distribution Mapping Map any X into Z-distribution.       -3 -2 -1 +1 +2 +3 0 Z X ~ N ( , 2 ) Z ~ N (0,1 2 ) Z = X -
Lec 10 - Normal Distribution IOE 265 F11 5 9 pdf and cdf of Z (standard normal random variable) Use tables or software to evaluate the cdf Devore Table A.3 (page 740) Excel =normsdist(z)

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ioe265f11-Lec10 - Lec 10 Normal Distribution Topics I...

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