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 braceleftbigg ( x μ ) 2 2 σ 2 bracerightbigg , − ∞ < 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 21.064 23.685 26.119 29.141 31.319 36.123 15 22.307 24.996 27.488 30.578 32.801 37.697 16 23.542 26.296 28.845 32.000 34.267 39.252 17 24.769 27.587 30.191 33.409 35.718 40.790 18 25.989 28.869 31.526 34.805 37.156 42.312 19 27.204 30.144 32.852 36.191 38.582 43.820 20 28.412 31.410 34.170 37.566 39.997 45.315 21 29.615 32.671 35.479 38.932 41.401 46.797 22 30.813 33.924 36.781 40.289 42.796 48.268 23 32.007 35.172 38.076 41.638 44.181 49.728 24 33.196 36.415 39.364 42.980 45.559 51.179 25 34.382 37.652 40.646 44.314 46.928 52.620 26 35.563 38.885 41.923 45.642 48.290 54.052 27 36.741 40.113 43.195 46.963 49.645 55.476 28 37.916 41.337 44.461 48.278 50.993 56.892 29 39.087 42.557 45.722 49.588 52.336 58.301 30 40.256 43.773 46.979 50.892 53.672 59.703
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T 7 Hypothesis Testing CS 531, Fall 2007 Copyright © William C. Cheng a test of a statistical hypothesis is based on observed values of the r.v. that leads to the acceptance or rejection of the hypothesis A statistical hypothesis, denoted H 0 , is an assertion about a distribution of one or more r.v.
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  • Spring '08
  • Cheng
  • Normal Distribution, Statistical hypothesis testing, Statistical significance, William C. Cheng

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