2 CHAPTER 2. ) ( ) 2 () () Octave rand() octave:27> A=rand(10,1) A =
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1 Chapter (cf. ) (= ) 76, 86, 77, 88, 78, 83, 86, 77, 74, 79, 82, 79, 80, 81, 78, 78, 73, 78, 81, 86, 71, 80, 81, 88, 82, 80, 80, 70, 77, 81 10? () ( 1
2 2 CHAPTER 2. ) ( ) 2 () () Octave rand() octave:27> A=rand(10,1) A = rand() (0,1)
3 octave:29> B=ceil(rand(10,1)*10) B = ceil() octave:30> hist(b) ( ) n hist(rand(n,1)) n (hist(rand(100,1) ) n n
4 4 CHAPTER A = (a 1, a 2,...a n ) µ µ = (1/n) n a i (2.1) Octave mean() Octave:> mean(a) i=1 10 n n 10,100,1000,,,, 1 80 : :
5 A = [10, 11, 10, 11, 10, 10, 11, 9, 11, 9, 11, 9, 9, 9, 11, 11, 12, 11, 11, 10] B = [4, 6, 5, 5, 6, 6, 6, 4, 5, 5, 18, 17, 17, 16, 16, 16, 15, 15, 13, 11] Octave A mean(a) A B A = (a 1, a 2,...a n ) σ 2 µ σ 2 = 1 n n (a i µ) 2 (2.2) i=1 σ 2 = 1 n = 1 n = 1 n = 1 n n (a i µ) 2 i=1 n (a 2 i 2a i µ + µ 2 ) i=1 n (a 2 i ) 1 n n 2µ a i + 1 n n µ2 i=1 i=1 n (a 2 i ) µ 2 (2.3) i=1 2 Octave var() A,B A B
6 6 CHAPTER 2. ( ) n n () Octave 1 sum(round(rand(100,1))*2-1) : #dist1.m n=10000; A=zeros(n,1); for(i=1:n) A(i,1)=sum(round(rand(100,1))*2-1); endfor mean(a) var(a) dist1.m Octave:> source("dist1.m"); n hist(a) round() round(rand()) [0,n-1] round(rand()*n) ceil(), floor() ceil(rand(n,1)*6) (n ) n? ( ) ( n )
7 : /4 3/4 (1 1 4 ) (1 3 ) = = 1 4 1/4 + 1/4 2 1 (1 ( )) 1 2 = = 1 4 1/4 + 1/4 + 1/4 (1 ( )) = 1 4 /and /or 1
8 8 CHAPTER ( 100 ) ( ) ()??( : )
9 Octave hist() ( histogram) Octave hist() ( () ) (=) randn() Octave ; (Octave ) A= randn(50,1); hist(a) hist(a,100) B1= randn(10000,1)+2.4; B2= randn(10000,1); B=[B1;B2]; hist(b) hist(b,100) hist(b) 10 hist(b,1000) 100
10 10 CHAPTER rand() (0,1) Octave rand() /6 f(x) ( ) f(x) x f(x)dx (2.4) 1 x [a, b] b a f(x)dx (2.5) x x
11 [0,1] 0.3 x < 0.4 f(x) = E(X) V (X) µ, σ 2 µ = E(X) = σ 2 = V (X) = xf(x)dx (2.6) (x µ) 2 f(x)dx (2.7) σ = V (X) 2.13 σ 2 = = (x µ) 2 f(x)dx x 2 f(x)dx 2µ xf(x)dx + µ 2 f(x)dx = E(X 2 ) 2µE(X) + µ 2 = E(X 2 ) E(X) 2 (2.8) µ = E(X) f(x)dx = 1 X X X = 1, 2, 3, 4, 5, 6 f(x) = 1/6 X E(X) E(X) = 6 k f(k) k=1 = ( 1 6 1) + (1 6 2) + (1 6 3) + (1 6 4) + (1 6 5) + (1 6 6) = 1 6 ( ) = (2.9)
12 12 CHAPTER 2. 1,2,...,n E(X) = n (2.10) V (X) 2.13 X 2 (E(X) 2 ) n k=1 k 2 = 1 n(n + 1)(2n + 1) (2.11) 6 E(X 2 ) = 1 n n k=1 k 2 = 1 n 1 n(n + 1)(2n + 1) 6 = 1 (n + 1)(2n + 1) (2.12) 6 V (X) = E(X 2 ) (E(X)) 2 (2.13) = 1 6 (n + 1)(2n + 1) 1 (n + 1)2 4 = 1 12 (n2 1) (2.14) n = ( ) ( ) p q = (1 p) n k b(k;n,p) ( ) n b(k; n, p) = p k q n k (2.15) k
13 ( ) n n! = n C k = k k!(n k)! 2 n k Octave n=10; p=0.5; x=1:n; pd=binomial_pdf(x,n,p); plot(pd); 2 1 : n 1 n ( ) binomial cdf(x,n,p) binomial cdf() np npq Figure 2.1: 2 n=50, p=0.1,0.2, 0.3, 0.4, 0.5 5% %
14 0 14 CHAPTER n 2 n ( ) x (Poisson) λ p(x, λ) p(x, λ) = λx x! e λ (2.16) λ λ n (λ = np )2 λ x Octave poisson pdf(x,lambda),poisson cdf(x,lambda) Figure 2.2: =1,2,
15 N(µ, σ 2 ) µ σ 2 x f(x) = 1 (x µ)2 e 2σ 2 (2.17) 2πσ 2 n 2 x dens ity % % 95.45% µ 3σ Figure 2.3: 2
16 16 CHAPTER 2. ( ) normal pdf(x,m,v), normal cdf(x,m,v) X m,v? n n n=100 ( 1/100 ) ( 100 ) Octave randn() normal cdf(x,m,v) 0, 1 x=-1, 1.5, 2.0 % normal cdf(x,m,v) 0, 1 x= [0.3, 0.5] % normal cdf(x,m,v) 0, 1 x= [1.0, 1.2] % URL
17 Z Z = X µ σ (2.18) µ σ X σ Z % 70 (µ = 0, σ 2 = 1) ( ) normal cdf(x,m,v) 0, 1 x=40, 55, 70 % dens ity % % 95.45% Figure 2.4: N(0,1)
18 18 CHAPTER (Zipf) 1/x(x 1 ) x α X X 2. 85% 1 95% (85%) ( 3 30 x6 ) ()
19 X x( ) 2σ 0.95 ( % ) Z Z = (X µ) σ (2.19) X µ, σ Z [-2,2] X x µ, σ 95% 60 ( 100) 95 () ( )
20 20 CHAPTER 2. 2 I false positive, II false negative, I ( ) () 2 YES/NO
21 t t t 2 2 ( ) 2 () t t t (Gosset) t t t X σ t = X µ σ/ n (2.20) Z t t t Octave ( ) (95% )
22 22 CHAPTER 2. 2 (χ 2 ) F ( ) Octave
23 χ 2 t χ 2 ( 2 ) χ 2 2 χ 2 2 A( :a 0, a 1,...a k, ( )n) P( :p 0, p 1,..., p k ) χ 2 (k-1) χ 2 χ 2 χ 2 = n (a i np i ) 2 i=1 np i (2.21) χ 2 B ( ) (B ) ˆp i = a i + b i n a + n b (2.22) n a n b A,B P [1/6,1/6,1/6,1/6,1/6,1/6] Octave χ 2 χ 2 Octave p (1-chisquare cdf(x,k)) X k chisq.m #chisq.m function x = chisq(a) length=size(a)(1,2); t0=round(sum(a)/length); x=0; for(i=1:length) x=x+ (A(1,i)-t0)^2/t0; endfor endfunction
24 24 CHAPTER 2. function p = dochisqtest(a) k=size(a)(1,2)-1; p = 1-chisquare_cdf(chisq(A),k); endfunction D1,D2,D3 D1=[11,20,9,16,19,25]; D2=[26,38,22,34,37,43]; D3=[44,52,39,56,53,56]; χ 2 Octave:134> source("chisq.m"); octave:135> chisq(d1) ans = octave:141> chisq(d2) ans = octave:142> chisq(d3) ans = k=5 95% octave:138> chisquare_inv(0.95,5) ans = χ (P: ) octave:143> dochisqtest(d1) ans = octave:144> dochisqtest(d2) ans = octave:145> dochisqtest(d3) ans = p 0.05 p 0.05 (p<0.05)
25 : Octave t t test(x,m,alt), t test 2(x,y,alt) p-value x,y ( ) m alt <> ( ) µ x < µ y < µ x > µ y > A1=randn(100,1)+10; A2=randn(200,1); 2 octave:173> t_test(a1,10) pval: ans = octave:174> t_test(a1,9) pval: 0 ans = 0 octave:175> t_test(a1,9) pval: 0 ans = 0 octave:176> t_test(a1,11) pval: e-19 ans = e-19 2 octave:177> t_test_2(a1,a2) pval: 0 ans = 0 octave:178> t_test_2(a1,a2,">") pval: 0 ans = 0 octave:179> t_test_2(a1,a2,"<") pval: 1 ans = 1 octave:180> mean(a1) ans =
26 26 CHAPTER 2. octave:181> mean(a2) ans = octave:183> A3=randn(10,1)+4; octave:184> t_test(a3,3) pval: ans = octave:185> t_test(a3,3.5) pval: ans = octave:186> t_test(a3,3.8) pval: ans = octave:187> t_test(a3,4.1) pval: ans = octave:188> t_test(a3,4.3) pval: ans = octave:189> t_test(a3,4.5) pval: ans = octave:190> t_test(a3,4.6) pval: ans = octave:191> t_test(a3,5) pval: ans = URL t t
27 : S.D. S.J.
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