1 c Stat Concept 1.0 a=0,b=4 0.8 0.6 0.4 0.2 Lognormal(a,b) a=0,b=.25 a=0,b=1 a=1,b=.25 a=1,b=1 1.0 0.8 0.6 0.4 0.2 Gamma(alpha) alpha=1 alpha=2 alpha=3 alpha=4 1.6 1.2 0.8 0.4 c=1 Weibull(c) c=4 c=3 c=2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 0.5 1.0 1.5 2.0 2.5 3.0 1.50 1.25 1.00 0.75 0.50 0.25 Laplace(mu,sig2) mu=0,sig2=.25 mu=0,sig2=1 mu=-2,sig2=1 mu=2,sig2=1 mu=0,sig2=4 0.8 0.6 0.4 0.2 Normal(mu,sig2) N(0,4) N(0,.25) N(-1,1) N(0,1)N(1,1) 0.6 0.5 0.4 0.3 0.2 0.1 Logistic(a,b) a=0,b=.5 a=-2,b=1 a=0,b=1 a=2,b=1 a=0,b=2-6 -4-2 0 1 2 3 4 5 6-5 -3-1 0 1 2 3 4 5-10 -6-2 0 2 4 6 8 10 Beta(p,q) 7 6 Beta(p=.5,q=.5) 5 4 Beta(p=2,q=9) Beta(p=9,q=2) 3 2 Beta(3,3) 1 0 0.2 0.4 0.6 0.8 1.0 Uniform(a,b) 2.5 2.0 U(-.25,.25) 1.5 1.0 0.5 U(-.75,75) U(0,1) U(-2,2) -2.00-1.00 0.75 1.75 Pareto(a) 4 a=4 3 a=3 2 a=2 1 a=1 0 1.0 1.2 1.4 1.6 1.8 2.0
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8. Stat Concept ( http://www.src.ac.ir E-mail: src@src.ac.ir E-mail: adel@aut.ac.ir E-mail: mohammadpour_adel@yahoo.com b, f.....,..! 52,
1 : " '.... 2 1, 4 3. f. 5 ) 8 ( 7 ) 6,. ( N. N. x N+1 = x 1 x N+2 = x 2. f. 9. 9 Probability 1 Relative Frequency 2 Observations 3 Random Variables 4 Distributions 5 Simulation 6 Statistical Simulation 7 Psedo Random Number 8 Seed 9
.1 10 f ( ). ( ) Time f. 10 Introduction to Concept Labs,.. :. f ",.,..,.. 10
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2 1.2,. 1 4, 3 2. 2.2 13 Sample 1 Independent 2 Identicall 3 Independence 4
.2 14? 6 5 (1 7 '. " ' "? 8 (2.? (3., (4?? (5?? (6? (7. 3 (8 1.2 (9? 5 (1.2 ) 9 Random Sampling " (10. Population 5 Finite 6 Simple Random Sample 7 Innite 8 9
15.2.2 Random Sampling Yellow boxes contain average of 5 values from horizontal axis corresponding to most recently selected individuals. Red indices inside boxes represent labels of individuals in the population. 10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15 14 13 12 11 30 29 28 27 26 25 24 23 22 21 40 39 38 37 36 35 34 33 32 31 50 49 48 47 46 45 44 43 42 41 0.51 60 59 58 57 56 55 54 53 52 51 70 69 68 67 66 65 64 63 62 61 80 79 78 77 76 75 74 73 72 71 90 89 88 87 86 85 84 83 82 81 100 99 98 97 96 95 94 93 92 91 5 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95.0/96,0/95,0/15,0/05 :1.2 ( 11 ) 10? " (11?? b " (12 25,Random Sampling : ) (., 12 " (13.? b " (14? (15., (16 Pribability MassFunction 10 Density 11 Relative Frequence Table 12
.2 16 ndraws = 1, npairs = 1, relative frequency = 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 :2.2 50? (2.2 ) 13 Relative Frequency and Probability " (17. 13
17.2.2 Theoretical (blue) and Observed (red) Histograms for a Discrete Probability Distribution (Chi-sq = 14.00 p-value = 0.1915 ) 28 2.80 3.00 3 56 5.60 5.00 5 83 8.30 8.00 8 0.11 11.00 9.00 9 0.14 14.00 6.00 6 0.17 17.00 16.00 0.16 0.14 14.00 18.00 0.18 0.11 11.00 12.00 0.12 83 8.30 16.00 0.16 56 5.60 5.00 5 28 2.80 2.00 2 0.20 0.15 0.10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 :3.2 (3.2 ) 14 Sampling from Discrete Populations (18. 14
.2 18 0.30 Sampling With (blue) and Without (red) Replacement (N = 50, n = 10 pi = 0.5 ) 0.25 0.20 0.15 0.10 5-1 0 1 2 3 4 5 6 7 8 9 10 11 " ' " ' :4.2 (19 Sampling from 0-1 Population > Sampling With and Without Replacment 16 15 (4.2 ) ( 18 17 ). :?! :! With Replacement 15 Without Replacement 16 Binomial 17 Hypergeometric 18
3 1.3,,.,,,. 1 ( 2 ),,, 3., 19 Empirical Density Function 1 Histogram 2 Normal 3
.3 20.. 2.3 (1.3 ) 4 How arepopulation Distributed (1 f.? " (2 ) Lab Option Simulated Data,, ( ).(2.3?. 4
21.2.3 0.8 Five Members of the Normal Family 0.7 0.6 N(0,.25) 0.5 0.4 N(-1,1) N(0,1) N(1,1) 0.3 0.2 0.1 N(0,4) -5-4 -3-2 -1 0 1 2 3 4 5 :1.3 Simulated Data from Normal (mean = 100, variance = 225) (Any data outside x range are in end boxes) 3 m1 = 100.9000 m2 = 227.5000 m3 = -28.1300 m4 = 1.5890e5 Skewness = -08196 Kurtosis = 3.0700 2 1 50 55 60 65 70 75 80 85 90 95 105 115 125 135 145 :2.3
.3 22 " (3., " (4, ( ).., " (5 ( ). Lab Option Simulated Data " (6. 500.( )?? (3.3 ) 5 Z, t, Chi-Square, F (7.,..,, f. ( )...!!... 5
23.2.3 Chi Square Curves 0.10 10 8 20 6 30 4 40 50 60 70 80 90 100 2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 :3.3
.3 24
4 1.4 1.. 2.,,. 3,. 5 4 25 Statistic 1 Parameter 2 Sampling Distribution 3 Weak Law of Large Number 4 Central Limit Theorem 5
.4 26 2.4 p 6 2 X 2 X 1 (1 X 2 X 1 x 2 x 1? T = X 1 + X 2? p t = x 1 + x 2 2 X 2 X 1 (2? X = 1 2 (X 1 + X 2 ) 2? 2 Random Sampling (3. 7? ( 20 f ) ) 2 2. (...., )? (., 5 f " (4? b 3 (5. 3. Bernoli 6 Mean 7
27.2.4. b.. 20 30 Random Sampling (6 5 2. 4 3 b ( ) 1 : " (7 X.,... X.,.... (8., (1.4 ) 8 Sampling Distributions (9 Z = X q ; E(X) V (X) E(1) U(0 1),N(0 1) ( ) 500,n = 20 f n. z = 20 x; 8
.4 28 0.50 0.45 0.40 0.35 Theoretical Distribution and Histogram of 500 Z Statistics Sampling from Normal(0,1), n = 5 Refer to the Report Window for numerical comparisons of the theoretical and Monte Carlo distributions. 0.30 0.25 0.20 0.15 0.10 5-3.50-2.80-2.10-1.40-0.70 0.70 1.40 2.10 2.80 3.50 Z :1.4 ).?,(?.?? (10.,(2.4 ) 9 Central Limit Theorom! (11 )? 9
29.2.4 3.0 Histogram of Sample Means of 500 Samples of Size 10 From Normal(0,1) Parent Population With Normal Theory Curve Superimposed 2.7 2.4 2.1 1.8 1.5 1.2 0.9 0.6 0.3-1.50-1.20-0.90-0.60-0.30 0.30 0.60 0.90 1.20 1.50 X :2.4.( b? Z (12. (13. (14.? (15 ". X n '? (16 n?,n (17? 30 f
.4 30? ( ) (18 ( b 20 19 b ).?. (19. P (0=001) 10 X 1 X 100 (a) P ( P 100 i=1 X i > 0). P (2) Y (b). P (Y > 0)?. (20. Statistical Tables. B(1 0=1) X 1 X 10 (a). P ( P 10 i=1 X i 2) B(1 0=3) X 1 X 10 (b). P ( P 10 i=1 X i 2). B(1 0=5) X 1 X 10 (c). P ( P 10 i=1 X i 2) (21 Sampling from 0- )??,(3.4 ) 1Population > Approximating Binomial Probabilities (.,." ' :!?, :? p X, X 2 n :,j n j' :, p X n "! Poisson 10
31.2.4 Poisson Approximation (red) to Binomial (blue) n = 50, pi = 0.1 0.15 0.10 5-1 0 1 2 3 4 5 6 7 8 9 10 11 12 0.30 P(X= 4 ): Approximating B(n,pi) by N(n*pi, n*pi*(1-pi)) n = 10, pi = 0.5 0.25 By bar: 0.2051 Under curve: 0.2045 0.20 0.15 0.10 5-1 0 1 2 3 4 5 6 7 8 9 10 11 :3.4