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slides10 - CS 531, Fall 2007 Normal Distribution The normal...

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T 1 Normal Distribution CS 531, Fall 2007 Copyright © William C. Cheng X is said to be N( μ , σ 2 ) A continuous r.v. X has a normal distribution with mean μ and variance σ 2 if its pdf is defined by: f ( x ) = 1 σ 2 π exp b ( x μ ) 2 2 σ 2 B , − ∞ < x < N(0,1) is called a standard normal distribution if the X is N( μ , σ 2 ) , then Z=(X- μ )/ σ is N(0,1) The normal distribution arises in practice when a large number of independent r.v. having the same mean and variance are summed Central Limit Theorem
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T 2 Normal Distribution (Cont. ..) CS 531, Fall 2007 Copyright © William C. Cheng f(x) for N(0,1) -3 -2 -1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 N(0,1) is symmetric about the vertical axis P(X > x) = P(X < -x) for any x
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T 3 Normal Distribution (Cont. ..) CS 531, Fall 2007 Copyright © William C. Cheng some values of P(X > x) = α for N(0,1) α x 0.1 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005 1.2816 1.6449 1.9600 2.3263 2.5758 2.8070 3.0902 3.2905 f(x) for N(0,1) -3 -2 -1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 5%
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A continuous r.v. X has a χ 2 distribution with v degrees of freedom ( v 1 ) if its pdf is defined by: where Γ is the gamma function The chi-square distribution can be used to compare the goodness-of-fit of the observed frequencies of events to their expected frequencies under a hypothesized distribution f ( x ) = 1 Γ( v/ 2)2 v/ 2 x ( v/ 2) 1 e x/ 2 if 0 x < 0 otherwise T 4 The Chi-square Distribution CS 531, Fall 2007 Copyright © William C. Cheng The chi-square distribution with v degrees of freedom arises in practice when the squares of v independent r.v. having standard normal distributions are summed mean and variance are μ = v and σ 2 = 2v
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0 5 10 15 20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T 5 The Chi-square Distribution (Cont. ..) CS 531, Fall 2007 Copyright © William C. Cheng f(x) for χ 2 distribution with v = 7 degrees of freedom if the X is N( μ , σ 2 ) , and σ 2 > 0, then Z=(X- μ ) 2 / σ 2 has a chi-square distribution with 1 degree of freedom if the X is N(0,1) , then Z=X 2 has a χ 2 distribution with 1 degree of freedom
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T 6 The Chi-square Distribution (Cont. ..) CS 531, Fall 2007 Copyright © William C. Cheng some values of P(X > x) = α for χ 2 distribution α v 0.1000 0.0500 0.0250 0.0100 0.0050 0.0010 -- --------------------------------------------------- 1 2.706 3.841 5.024 6.635 7.879 10.828 2 4.605 5.991 7.378 9.210 10.597 13.816 3 6.251 7.815 9.348 11.345 12.838 16.266 4 7.779 9.488 11.143 13.277 14.860 18.467 5 9.236 11.070 12.833 15.086 16.750 20.515 6 10.645 12.592 14.449 16.812 18.548 22.458 7 12.017 14.067 16.013 18.475 20.278 24.322 8 13.362 15.507 17.535 20.090 21.955 26.124 9 14.684 16.919 19.023 21.666 23.589 27.877 10 15.987 18.307 20.483 23.209 25.188 29.588 11 17.275 19.675 21.920 24.725 26.757 31.264 12 18.549 21.026 23.337 26.217 28.300 32.909 13 19.812 22.362 24.736 27.688 29.819 34.528 14
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slides10 - CS 531, Fall 2007 Normal Distribution The normal...

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