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# Lec08 - 3-5 Hypergeometric Distribution IOE/Stat 265 Fall...

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1 Ch. 3.5-3.6 IOE/Stat 265, Fall 2009 IOE/Stat 265, Fall 2009 Lecture #8: Lecture #8: Poisson Distribution (and Poisson Distribution (and Hypergeometric Hypergeometric ) 2 3-5 Hypergeometric Distribution ± For a finite population of N items of which M are “successes", a sample of n independent observations is drawn without replacement. The exact probability, p(x), that the sample will contain exactly x defectives is given by x = max{0,n-N+M} to min(n,M) ⎛⎞ ⎜⎟ ⎝⎠ = MNM xnx p(x) N n 3 Mean and Variance == nM M E(X) np for p= NN ± ± ± EXCEL: = HYPGEOMDIST( x , n , M , N) =− Nnn M M V(X) 1 N1 N N Nn np(1 p) N1 Finite Population Correction Factor 4 Example 1: Circuit Board Testing ± Special-purpose circuit boards are produced in lots of size N = 20. The boards are accepted in a sample of n = 3 if all are conforming. The entire sample is drawn from the lot at one time and tested. If the lot contains M = 3 nonconforming boards, what is the probability of acceptance?

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6 Approximating the Hypergeometric ± Rule of Thumb: Define p = M/N, For N>100 and n/N < 0.05 Binomial Distribution provides a reasonable approximation to the Hypergeometric Distribution 7 Using Minitab to Compute Probabilities Let n = 10 Hypergeometric Finite Population (N 100 and n/N 0.05) Binomial Large Population (N 100 and n/N 0.05) Poisson Approximation (np 5, p 0.10 or p 0.90) Normal Approximation (np 5 and 0.10 p 0.90) APPROXIMATIONS 9 3-6 Poisson Distribution λ = rate per unit time or unit area x = 0,1,2, … where x is the number of defects * Law of small numbers: events with low frequency in a large population follow a Poisson Distribution −λ λ = λ> x e p(x) x! for some 0 PMF: von Bortkiewicz Distribution?
10 Cumulative Poisson Table ~ F(x, λ ) x 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 0 0.135 0.050 0.018 0.007 0.002 0.001 0.000 0.000 0.000 0.000 0.000 1 0.406 0.199 0.092 0.040 0.017 0.007 0.003 0.001 0.000 0.000 0.000 2 0.677 0.423 0.238 0.125 0.062 0.030 0.014 0.006 0.003 0.000 0.000 3 0.857 0.647 0.433 0.265 0.151 0.082 0.042 0.021 0.010 0.000 0.000 4 0.947 0.815 0.629 0.440 0.285 0.173 0.100 0.055 0.029 0.001 0.000 5 0.983 0.916 0.785 0.616 0.446 0.301 0.191 0.116 0.067 0.003 0.000 6 0.995 0.966 0.889 0.762 0.606 0.450 0.313 0.207 0.130 0.008 0.000 7 0.999 0.988 0.949 0.867 0.744 0.599 0.453 0.324 0.220 0.018 0.001 8 1.000 0.996 0.979 0.932 0.847 0.729 0.593 0.456 0.333 0.037 0.002 9 0.999 0.992 0.968 0.916 0.830 0.717 0.587 0.458 0.070 0.005 10 1.000 0.997 0.986 0.957 0.901 0.816 0.706 0.583 0.118 0.011 11 0.999 0.995 0.980 0.947 0.888 0.803 0.697 0.185 0.021 12 1.000 0.998 0.991 0.973 0.936 0.876 0.792 0.268 0.039 13 0.999 0.996 0.987 0.966 0.926 0.864 0.363 0.066 14 1.000 0.999 0.994 0.983 0.959 0.917 0.466 0.105 15 0.999 0.998 0.992 0.978 0.951 0.568 0.157 16 1.000 0.999 0.996 0.989 0.973 0.664 0.221 17 1.000 0.998 0.995 0.986 0.749 0.297 18 0.999 0.998 0.993 0.819 0.381 19 1.000 0.999 0.997 0.875 0.470 20 1.000 0.998 0.917 0.559 21 0.999 0.947 0.644 22 1.000 0.967 0.721 36 1.000 λ APPENDIX A.2

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Lec08 - 3-5 Hypergeometric Distribution IOE/Stat 265 Fall...

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