Lec10 - IOE/Stat IOE/Stat 265, Fall 2009 Lecture #10:...

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IOE/Stat 265, Fall 2009 Lecture #10: ectu e 0 (Uniform), Normal, and Exponential Distributions po a s bu o s Ch 4-3 – 4-4 1 ontinuous Uniform Distribution Continuous Uniform Distribution random variable X has a continuous uniform ± A random variable X has a continuous uniform distribution if f(x) = 1/(b - a) for a x b 1 a x b f(x) (b a) 0 otherwise ≤≤ = Distribution Plot Uniform, Lower=5, Upper=11 ab (x) + = = 0.18 0.16 0.14 0.12 y 2 E(x) 2 μ= 0.10 0.08 0.06 0.04 Densit 2 ( ) 22 b a E(x ) 12 σ= −μ = 11 10 9 8 7 6 5 0.02 0.00 X DF for Continuous Uniform Distribution CDF for Continuous Uniform Distribution () 0 xa F(X x) a x b a < == ( ) ba xb 1 > CDF: F(x) vs x 1.0 0.8 0.6 0.4 0.2 F(x) 3 16 14 12 10 8 6 4 2 0 0.0 x Normal Distribution Properties 4-3 Normal Distribution Properties he normal distribution is undoubtedly the most ± The normal distribution is undoubtedly the most important and useful one of them all! ± Mean, Median, Mode all coincide. ± Symmetrical. Distribution Plot Normal, Mean=10, StDev=3 Notation: X ~ N ( μ , σ 2 ) 0.14 0.12 0.10 0.08 0.06 0.04 Density σ 4 μ 20 15 10 5 0 0.02 0.00 X μ
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pplications Applications ormal Distributions are approximately EVERYWHERE! ± Normal Distributions are approximately EVERYWHERE! ± Examples: Scores ± 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 arenas often tries to figure out why data are NOT normally distributed. 5 robability Distributions Probability Distributions robability Density Function for Normal ± Probability Density Function for Normal 22 () / ( 2 ) 1 ; ) - x μσ −− < < (; , 2 fx e πσ =∞ and 0 −∞ < μ < ∞ σ > ± Example: ± Suppose X~N(100,1 2 ), find f(X=100) Æ X N(100 5 fi d f(X 100) ± Suppose X~N(100,5 2 ), find f(X=100) Æ ± Suppose X~N(100,10 2 ), find f(X=100) Æ 6 ± Notice the scaling effect ~ remember condition 2 of pdf 0.4 1 StDev Distribution Plot Normal, Mean=100 0.3 ty 5 10 0.2 0.1 Densi 120 110 100 90 80 0.0 8 X tandardizing a Variable Standardizing a Variable o simplify the integration of f(x we will map ± To simplify the integration of f(x, μ , σ ), we will map our data into Standardized Form, aka “Z-score”. ± Z-score standardizes a variable, X, so that Z has a mean = 0, and variance = 1. ± We may standardize our variable, X, by performing the following Z transform: x z −μ = σ 9
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ormal Distribution Mapping Normal Distribution Mapping ± Map any X into Z distribution. ~ 2 −3σ −2σ −1σ +1σ +2σ +3σ μ X N ( μ , σ ) X − μ -3 -2 -1 +1 +2 +3 0 Z Z ~ N (0,1 2 ) μ = σ x z 10 PDF and CDF of Z (Standard ormal Random Variable) Normal Random Variable) 2 z/2 1 f(z;0,1) e - z =∞ < < PDF: 2 π CDF: (z) P(Z z) ϕ= se Tables or Software to evaluate the CDF(z): ± Use Tables or Software to evaluate the CDF(z): ± Appendix A3 (page 668) ± Excel =normsdist(z) or =normdist(x, μ , σ ,true) 11 ± Minitab Calc >>Probability Distributions >>Normal ormal Table Symmetry Normal Table Symmetry seful Equations: ± Useful Equations: ± P(Z > z) = 1 – P(Z z) ± P(Z > z) = P(Z -z) ± Do we need positive and negative Z-tables?
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This note was uploaded on 03/17/2010 for the course IOE 265 taught by Professor Garyherrin during the Fall '09 term at University of Michigan-Dearborn.

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Lec10 - IOE/Stat IOE/Stat 265, Fall 2009 Lecture #10:...

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