Probability ad Distributios A Brief Itroductio
Radom Variables Radom Variable (RV): A umeric outcome that results from a experimet For each elemet of a experimet s sample space, the radom variable ca take o exactly oe value Discrete Radom Variable: A RV that ca take o oly a fiite or coutably ifiite set of outcomes Cotiuous Radom Variable: A RV that ca take o ay value alog a cotiuum (but may be reported discretely ) Radom Variables are deoted by upper case letters (Y) Idividual outcomes for RV are deoted by lower case letters (y)
Probability Distributios Probability Distributio: Table, Graph, or Formula that describes values a radom variable ca take o, ad its correspodig probability (discrete RV) or desity (cotiuous RV) Discrete Probability Distributio: Assigs probabilities (masses) to the idividual outcomes Cotiuous Probability Distributio: Assigs desity at idividual poits, probability of rages ca be obtaied by itegratig desity fuctio Discrete Probabilities deoted by: p(y) = P(Y=y) Cotiuous Desities deoted by: f(y) Cumulative Distributio Fuctio: F(y) = P(Y y)
Discrete Probability Distributios Probability (Mass) Fuctio: p( y) P( Y y) p( y) 0 y all y py ( ) 1 Cumulative Distributio Fuctio (CDF): F( y) P( Y y) F( b) P( Y b) p( y) F( ) 0 F( ) 1 F( y) is mootoically icreasig i y b y
Cotiuous Radom Variables ad Probability Distributios Radom Variable: Y Cumulative Distributio Fuctio (CDF): F(y)=P(Y y) Probability Desity Fuctio (pdf): f(y)=df(y)/dy Rules goverig cotiuous distributios: f(y) 0 y f ( y) dy 1 P(a Y b) = F(b)-F(a) = b f ( y ) dy a P(Y=a) = 0 a
Expected Values of Cotiuous RVs Expected Value: E( Y ) yf ( y) dy (assumig absolute covergece) E g( Y ) g( y) f ( y) dy Variace: ( ) ( ( )) ( ) ( ) V Y E Y E Y y f y dy y y f ( y) dy y f ( y) dy yf ( y) dy f ( y) dy E Y ( ) (1) E Y E ay b ( ay b) f ( y) dy a yf ( y) dy b f ( y) dy a( ) b(1) a b V ay b E ( ay b) E( ay b) ( ay b) ( a b) f ( y) dy ay b ( ay a) f ( y) dy a ( y ) f ( y) dy a V ( Y ) a a
Meas ad Variaces of Liear Fuctios of RVs i1 U a Y a costats Y radom variables i i i i, E Yi i V Yi i COV Yi Yj E Yi i Yj j ij E U E aiyi aii i1 i1 V U V a Y a a a 1 i i i i i jij i1 i1 i1 ji1 Y1,..., Y idepedet V U V a Y i i i 1 i1 a i i
Normal (Gaussia) Distributio Bell-shaped distributio with tedecy for idividuals to clump aroud the group media/mea Used to model may biological pheomea May estimators have approximate ormal samplig distributios (see Cetral Limit Theorem) Notatio: Y~N(, ) where is mea ad is variace 1 ( y) 1 f y e y ( ),, 0 Obtaiig Probabilities i EXCEL: To obtai: F(y)=P(Y y) Use Fuctio: =NORMDIST(y,,,1) Virtually all statistics textbooks give the cdf (or upper tail probabilities) for stadardized ormal radom variables: z=(y-)/ ~ N(0,1)
f(y) Normal Distributio Desity Fuctios (pdf) Normal Desities 0.045 0.04 0.035 0.03 0.05 0.0 N(100,400) N(100,100) N(100,900) N(75,400) N(15,400) 0.015 0.01 0.005 0 0 0 40 60 80 100 10 140 160 180 00 y
Secod Decimal Place of z Iteger part ad first decimal place of z 1-F(z) 0.00 0.01 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.4960 0.490 0.4880 0.4840 0.4801 0.4761 0.471 0.4681 0.4641 0.1 0.460 0.456 0.45 0.4483 0.4443 0.4404 0.4364 0.435 0.486 0.447 0. 0.407 0.4168 0.419 0.4090 0.405 0.4013 0.3974 0.3936 0.3897 0.3859 0.3 0.381 0.3783 0.3745 0.3707 0.3669 0.363 0.3594 0.3557 0.350 0.3483 0.4 0.3446 0.3409 0.337 0.3336 0.3300 0.364 0.38 0.319 0.3156 0.311 0.5 0.3085 0.3050 0.3015 0.981 0.946 0.91 0.877 0.843 0.810 0.776 0.6 0.743 0.709 0.676 0.643 0.611 0.578 0.546 0.514 0.483 0.451 0.7 0.40 0.389 0.358 0.37 0.96 0.66 0.36 0.06 0.177 0.148 0.8 0.119 0.090 0.061 0.033 0.005 0.1977 0.1949 0.19 0.1894 0.1867 0.9 0.1841 0.1814 0.1788 0.176 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 1.0 0.1587 0.156 0.1539 0.1515 0.149 0.1469 0.1446 0.143 0.1401 0.1379 1.1 0.1357 0.1335 0.1314 0.19 0.171 0.151 0.130 0.110 0.1190 0.1170 1. 0.1151 0.1131 0.111 0.1093 0.1075 0.1056 0.1038 0.100 0.1003 0.0985 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.083 1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.071 0.0708 0.0694 0.0681 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.058 0.0571 0.0559 1.6 0.0548 0.0537 0.056 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 1.7 0.0446 0.0436 0.047 0.0418 0.0409 0.0401 0.039 0.0384 0.0375 0.0367 1.8 0.0359 0.0351 0.0344 0.0336 0.039 0.03 0.0314 0.0307 0.0301 0.094 1.9 0.087 0.081 0.074 0.068 0.06 0.056 0.050 0.044 0.039 0.033.0 0.08 0.0 0.017 0.01 0.007 0.00 0.0197 0.019 0.0188 0.0183.1 0.0179 0.0174 0.0170 0.0166 0.016 0.0158 0.0154 0.0150 0.0146 0.0143. 0.0139 0.0136 0.013 0.019 0.015 0.01 0.0119 0.0116 0.0113 0.0110.3 0.0107 0.0104 0.010 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084.4 0.008 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064.5 0.006 0.0060 0.0059 0.0057 0.0055 0.0054 0.005 0.0051 0.0049 0.0048.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036.7 0.0035 0.0034 0.0033 0.003 0.0031 0.0030 0.009 0.008 0.007 0.006.8 0.006 0.005 0.004 0.003 0.003 0.00 0.001 0.001 0.000 0.0019.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 3.0 0.0013 0.0013 0.0013 0.001 0.001 0.0011 0.0011 0.0011 0.0010 0.0010
Chi-Square Distributio Idexed by degrees of freedom () X~c Z~N(0,1) Z ~c 1 Assumig Idepedece: X,..., X ~ c i 1,..., X ~ 1 Desity Fuctio: i i c i i1 1 0, 0 1 x f x x e x Obtaiig Probabilities i EXCEL: To obtai: 1-F(x)=P(X x) Use Fuctio: =CHIDIST(x,) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper (ad sometimes lower) tail probabilities
f(x^) Chi-Square Distributios Chi-Square Distributios 0. 0.18 df=4 0.16 0.14 df=10 0.1 0.1 0.08 df=0 df=30 df=50 f1(y) f(y) f3(y) f4(y) f5(y) 0.06 0.04 0.0 0 0 10 0 30 40 50 60 70 X^
Critical Values for Chi-Square Distributios (Mea=, Variace=) df\f(x) 0.005 0.01 0.05 0.05 0.1 0.9 0.95 0.975 0.99 0.995 1 0.000 0.000 0.001 0.004 0.016.706 3.841 5.04 6.635 7.879 0.010 0.00 0.051 0.103 0.11 4.605 5.991 7.378 9.10 10.597 3 0.07 0.115 0.16 0.35 0.584 6.51 7.815 9.348 11.345 1.838 4 0.07 0.97 0.484 0.711 1.064 7.779 9.488 11.143 13.77 14.860 5 0.41 0.554 0.831 1.145 1.610 9.36 11.070 1.833 15.086 16.750 6 0.676 0.87 1.37 1.635.04 10.645 1.59 14.449 16.81 18.548 7 0.989 1.39 1.690.167.833 1.017 14.067 16.013 18.475 0.78 8 1.344 1.646.180.733 3.490 13.36 15.507 17.535 0.090 1.955 9 1.735.088.700 3.35 4.168 14.684 16.919 19.03 1.666 3.589 10.156.558 3.47 3.940 4.865 15.987 18.307 0.483 3.09 5.188 11.603 3.053 3.816 4.575 5.578 17.75 19.675 1.90 4.75 6.757 1 3.074 3.571 4.404 5.6 6.304 18.549 1.06 3.337 6.17 8.300 13 3.565 4.107 5.009 5.89 7.04 19.81.36 4.736 7.688 9.819 14 4.075 4.660 5.69 6.571 7.790 1.064 3.685 6.119 9.141 31.319 15 4.601 5.9 6.6 7.61 8.547.307 4.996 7.488 30.578 3.801 16 5.14 5.81 6.908 7.96 9.31 3.54 6.96 8.845 3.000 34.67 17 5.697 6.408 7.564 8.67 10.085 4.769 7.587 30.191 33.409 35.718 18 6.65 7.015 8.31 9.390 10.865 5.989 8.869 31.56 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 7.04 30.144 3.85 36.191 38.58 0 7.434 8.60 9.591 10.851 1.443 8.41 31.410 34.170 37.566 39.997 1 8.034 8.897 10.83 11.591 13.40 9.615 3.671 35.479 38.93 41.401 8.643 9.54 10.98 1.338 14.041 30.813 33.94 36.781 40.89 4.796 3 9.60 10.196 11.689 13.091 14.848 3.007 35.17 38.076 41.638 44.181 4 9.886 10.856 1.401 13.848 15.659 33.196 36.415 39.364 4.980 45.559 5 10.50 11.54 13.10 14.611 16.473 34.38 37.65 40.646 44.314 46.98 6 11.160 1.198 13.844 15.379 17.9 35.563 38.885 41.93 45.64 48.90 7 11.808 1.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645 8 1.461 13.565 15.308 16.98 18.939 37.916 41.337 44.461 48.78 50.993 9 13.11 14.56 16.047 17.708 19.768 39.087 4.557 45.7 49.588 5.336 30 13.787 14.953 16.791 18.493 0.599 40.56 43.773 46.979 50.89 53.67 40 0.707.164 4.433 6.509 9.051 51.805 55.758 59.34 63.691 66.766 50 7.991 9.707 3.357 34.764 37.689 63.167 67.505 71.40 76.154 79.490 60 35.534 37.485 40.48 43.188 46.459 74.397 79.08 83.98 88.379 91.95 70 43.75 45.44 48.758 51.739 55.39 85.57 90.531 95.03 100.45 104.15 80 51.17 53.540 57.153 60.391 64.78 96.578 101.879 106.69 11.39 116.31 90 59.196 61.754 65.647 69.16 73.91 107.565 113.145 118.136 14.116 18.99 100 67.38 70.065 74. 77.99 8.358 118.498 14.34 19.561 135.807 140.169
Studet s t-distributio Idexed by degrees of freedom () X~t Z~N(0,1), X~c Assumig Idepedece of Z ad X: T Desity Fuctio: f t Z ~ t X 1 1 t 1 t 0 Obtaiig Probabilities i EXCEL: To obtai: 1-F(t)=P(T t) Use Fuctio: =TDIST(t,) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper tail probabilities
Desity t(3), t(11), t(4), Z Distributios 0.45 0.4 0.35 0.3 0.5 0. 0.15 f(t_3) f(t_11) f(t_4) Z~N(0,1) 0.1 0.05 0-3 - -1 0 1 3 t (z)
Critical Values for Studet s t-distributios (Mea=, Variace=) df\f(t) 0.9 0.95 0.975 0.99 0.995 1 3.078 6.314 1.706 31.81 63.657 1.886.90 4.303 6.965 9.95 3 1.638.353 3.18 4.541 5.841 4 1.533.13.776 3.747 4.604 5 1.476.015.571 3.365 4.03 6 1.440 1.943.447 3.143 3.707 7 1.415 1.895.365.998 3.499 8 1.397 1.860.306.896 3.355 9 1.383 1.833.6.81 3.50 10 1.37 1.81.8.764 3.169 11 1.363 1.796.01.718 3.106 1 1.356 1.78.179.681 3.055 13 1.350 1.771.160.650 3.01 14 1.345 1.761.145.64.977 15 1.341 1.753.131.60.947 16 1.337 1.746.10.583.91 17 1.333 1.740.110.567.898 18 1.330 1.734.101.55.878 19 1.38 1.79.093.539.861 0 1.35 1.75.086.58.845 1 1.33 1.71.080.518.831 1.31 1.717.074.508.819 3 1.319 1.714.069.500.807 4 1.318 1.711.064.49.797 5 1.316 1.708.060.485.787 6 1.315 1.706.056.479.779 7 1.314 1.703.05.473.771 8 1.313 1.701.048.467.763 9 1.311 1.699.045.46.756 30 1.310 1.697.04.457.750 40 1.303 1.684.01.43.704 50 1.99 1.676.009.403.678 60 1.96 1.671.000.390.660 70 1.94 1.667 1.994.381.648 80 1.9 1.664 1.990.374.639 90 1.91 1.66 1.987.368.63 100 1.90 1.660 1.984.364.66
F-Distributio Idexed by degrees of freedom ( 1, ) W~F 1, X 1 ~c 1, X ~c Assumig Idepedece of X 1 ad X : W X X 1 1 ~ F, 1 Desity Fuctio: 1 1 1 1 1 w 1 1 1 1 1 f w w 1 w 0 1 1, 0 1 Obtaiig Probabilities i EXCEL: To obtai: 1-F(w)=P(W w) Use Fuctio: =FDIST(w, 1, ) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper tail probabilities
Desity Fuctio of F F-Distributios 0.9 0.8 0.7 0.6 0.5 0.4 f(5,5) f(5,10) f(10,0) 0.3 0. 0.1 0-0.1 0 1 3 4 5 6 7 8 9 10 F
Critical Values for F-distributios P(F Table Value) = 0.95 df\df1 1 3 4 5 6 7 8 9 10 1 161.45 199.50 15.71 4.58 30.16 33.99 36.77 38.88 40.54 41.88 18.51 19.00 19.16 19.5 19.30 19.33 19.35 19.37 19.38 19.40 3 10.13 9.55 9.8 9.1 9.01 8.94 8.89 8.85 8.81 8.79 4 7.71 6.94 6.59 6.39 6.6 6.16 6.09 6.04 6.00 5.96 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.8 4.77 4.74 6 5.99 5.14 4.76 4.53 4.39 4.8 4.1 4.15 4.10 4.06 7 5.59 4.74 4.35 4.1 3.97 3.87 3.79 3.73 3.68 3.64 8 5.3 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 9 5.1 4.6 3.86 3.63 3.48 3.37 3.9 3.3 3.18 3.14 10 4.96 4.10 3.71 3.48 3.33 3. 3.14 3.07 3.0.98 11 4.84 3.98 3.59 3.36 3.0 3.09 3.01.95.90.85 1 4.75 3.89 3.49 3.6 3.11 3.00.91.85.80.75 13 4.67 3.81 3.41 3.18 3.03.9.83.77.71.67 14 4.60 3.74 3.34 3.11.96.85.76.70.65.60 15 4.54 3.68 3.9 3.06.90.79.71.64.59.54 16 4.49 3.63 3.4 3.01.85.74.66.59.54.49 17 4.45 3.59 3.0.96.81.70.61.55.49.45 18 4.41 3.55 3.16.93.77.66.58.51.46.41 19 4.38 3.5 3.13.90.74.63.54.48.4.38 0 4.35 3.49 3.10.87.71.60.51.45.39.35 1 4.3 3.47 3.07.84.68.57.49.4.37.3 4.30 3.44 3.05.8.66.55.46.40.34.30 3 4.8 3.4 3.03.80.64.53.44.37.3.7 4 4.6 3.40 3.01.78.6.51.4.36.30.5 5 4.4 3.39.99.76.60.49.40.34.8.4 6 4.3 3.37.98.74.59.47.39.3.7. 7 4.1 3.35.96.73.57.46.37.31.5.0 8 4.0 3.34.95.71.56.45.36.9.4.19 9 4.18 3.33.93.70.55.43.35.8..18 30 4.17 3.3.9.69.53.4.33.7.1.16 40 4.08 3.3.84.61.45.34.5.18.1.08 50 4.03 3.18.79.56.40.9.0.13.07.03 60 4.00 3.15.76.53.37.5.17.10.04 1.99 70 3.98 3.13.74.50.35.3.14.07.0 1.97 80 3.96 3.11.7.49.33.1.13.06.00 1.95 90 3.95 3.10.71.47.3.0.11.04 1.99 1.94 100 3.94 3.09.70.46.31.19.10.03 1.97 1.93