数学カフェ 確率・統計・機械学習回 「確率・統計入門」

数学カフェ 確率・統計・機械学習回 「確率・統計入門」

1. 1. Twitter
2. 2. 990 10 583.7 170.1 http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html
3. 3. 990 10 583.7 170.1 http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html
4. 4. D = {x1, x2, · · · , xn} ¯x = 1 n nX i=1 xi 2 = 1 n nX i=1 (xi ¯x)2 = v u u t 1 n nX i=1 (xi ¯x)2
5. 5. = 1 n nX i=1 |xi ¯x|
6. 6. = 1 n nX i=1 (xi ¯x)2
7. 7. p
8. 8. = v u u t 1 N NX i=1 (xi ¯x)2 p
9. 9. probability
10. 10. ! 2 ⌦ = {!1, !2, · · · , !m} ⌦ = { , } ! 2 { , } !(1) = !(2) = !(n) =
11. 11. ⌦ = {1, 2, 3, 4, 5, 6} !(1) = !(2) = !(n) = ⌦ = {!1, !2, · · · , !49870000} !(1) = !43890298 = 171cm !(2) = !29184638 = 168cm !(n) = !51398579 = 174cm
12. 12. !(1) = !(2) = !(n) =!(3) = !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 = {!1, !2, !3, · · · , !10} ! 2 ⌦ = {ID1, ID2, ID3, · · · , ID10}
13. 13. ⌦ ! !
14. 14. X = X(!) ⌦ ! ! X(!1) = 0 X(!2) = 0 X(!3) = 0 X(!4) = 0 X(!5) = 0 X(!6) = 0 X(!7) = 0 X(!8) = 0 X(!9) = 0 X(!10) = 100
15. 15. ! {! 2 ⌦ : X(!) 2 A} {X 2 A} X(!) X
16. 16. {! 2 ⌦ : X(!) 2 A} !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 A X(!) = 100Ac X(!) = 0 !5 or !9
17. 17. PX (A) = P(X 2 A) = P({! 2 ⌦ : X(!) 2 A}) ⌦ !5, !9 !5, !9 PX (A) = #({! 2 ⌦ : X(!) 2 A}) #( ) = #(!5, !9) #( ) = 2 10 = 0.2
18. 18. PX(⌦) = 1 A1, A2, · · · PX ([iAi) = X i PX (Ai) A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 0  PX(A)  1
19. 19. X = X(!) ⌦ A A !1 !2 !3 !4 !5 !6 !7 !8 !11 !10 !9 !12 !13 !14 !15 !16 B C D X(!) = 0 X(!) = 0 #A = #{! 2 ⌦ : X(!) = 0} = 7 #B = #{! 2 ⌦ : X(!) = 1} = 2 #C = #{! 2 ⌦ : X(!) = 2} = 4 #D = #{! 2 ⌦ : X(!) = 3} = 3
20. 20. ⌦ A A !1 !2 !3 !4 !5 !6 !7 !8 !11 !10 !9 !12 !13 !14 !15 !16 B C DX(!) = 0 P(X = 0) = PX(A) = #{! 2 ⌦ : X(!) = 0} #⌦ = 7 16 P(X = 1) = PX (B) = #{! 2 ⌦ : X(!) = 1} #⌦ = 2 16 P(X = 2) = PX(C) = #{! 2 ⌦ : X(!) = 2} #⌦ = 4 16 P(X = 3) = PX(D) = #{! 2 ⌦ : X(!) = 3} #⌦ = 3 16
21. 21. {x1, x2, · · · , xk} P(X = xi) = f(xi) F(x) = P(X  x)
22. 22. P(x < X  x + x) x + xx x x ! 0 f(x) = lim x!0 P(x < X  x + x) x
23. 23. x + xx f(x) F(x) = P(X  x) = Z x 1 f(u)du f(a < x < b) = Z b a f(x)dx
24. 24. http://www.math.wm.edu/~leemis/2008amstat.pdf
25. 25. P(X = x) = px (1 p)1 x (x = 0, 1)
26. 26. # # p = 0.7 trial_size = 10000 set.seed(71) # data <- rbern(trial_size, p) # dens <- data.frame(y=c((1-p),p)*trial_size, x=c(0, 1)) # ggplot() + layer(data=data.frame(x=data), mapping=aes(x=x), geom="bar", stat="bin", bandwidth=0.1 ) + layer(data=dens, mapping=aes(x=x, y=y), geom="bar", stat="identity", width=0.05, fill="#777799", alpha=0.7)
27. 27. P(X = x) = nCrpx (1 p)n x (x = 0, 1, · · · , n)
28. 28. # p = 0.7 trial_size = 10000 sample_size = 30 set.seed(71) # gen_binom_var <- function() { return(sum(rbern(sample_size, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dbinom(seq(sample_size), sample_size, 0.7))*trial_size # ggplot() + layer(data=resuylt, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=1, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(sample_size)+.5, y=y), geom="line", stat="identity", position="identity",colour="red" ) + ggtitle("Bernoulli to Binomial.")
29. 29. P(X = x) = e x x! (x = 1, 2, · · · , 1)
30. 30. trial_size = 5000; width <- 1; # p = 0.7; n = 10; np <- p*n # n!∞ p!0 np= n = 100000; p <- np/n # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dpois(seq(20), np))*trial_size # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(20)+.5, y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Poisson.")
31. 31. f(x) = 1 p 2⇡ 2 exp ⇢ 1 2 (x µ)2 2 ( 1 < x < 1)
32. 32. # n <- 10000; p <- 0.7; trial_size = 10000 width=10 # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dnorm(seq(6800,7200), mean=n*p, sd=sqrt(n*p*(1-p)))*trial_size*width) # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(6800,7200), y=y), geom="line", stat="identity", position="identity", colour="red") + ggtitle("Bernoulli to Normal.")
33. 33. ( 1 < x < 1) f(x) = 1 p 2⇡ exp ⇢ 1 2 x2
34. 34. # n <- 10000; p <- 0.7 trial_size = 30000 width=0.18 # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) m <- mean(result$V1); sd <- sd(result$V1); result <- (result - m)/sd # dens <- data.frame(y=dnorm(seq(-4,4,0.05), mean=0, sd=1)*trial_size*width) # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(-4,4,0.05), y=y), geom="line", stat="identity", position=“identity", colour="red" ) + ggtitle("Bernoulli to Standard Normal.")
35. 35. f(x, k) = (1/2)k/2 (k/2) xk/2 1 e x/2 (0  x) Xi Z = X2 1 + · · · + X2 k
36. 36. # p <- 0.7; n <- 1000; trial_size <- 100000; width <- 0.3; df <- 3 # (3 ) gen_binom_var <- function() { return(sum(rbern(n, p))) } gen_chisq_var <- function() { result <- rdply(trial_size, gen_binom_var()) return(((result$V1 - mean(result$V1))/sd(result$V1))**2) } # df result <- rlply(df, gen_chisq_var(),.progress = "text") res <- data.frame(x=result[[1]] + result[[2]] + result[[3]]) # ( =3) xx <- seq(0,20,0.1) dens <- data.frame(y=dchisq(x=xx, df=df)*trial_size*width) # ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=xx, y=y), geom="line", stat="identity", position="identity", colour="blue" ) + ggtitle("Bernoulli to Chisquare")
37. 37. f(x, ) = ⇢ e x (x 0) 0 (x < 0)
38. 38. trial_size = 7000; width <- .01; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n # gen_exp_var <- function() { cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) # 1 } } } data <- data.frame(x=rdply(trial_size, gen_exp_var())/n) names(data) <- c("n", "x") # dens <- data.frame(y=dexp(seq(0, 1.5, 0.1), np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0, 1.5, 0.1), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Exponential.")
39. 39. f(x, ↵, ) = ↵ (↵) x↵ 1 exp( x) (0  x < 1) ↵X i=1 Xi ⇠ (↵, )Xi ⇠ Exp( )
40. 40. trial_size = 7000; width <- .035; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n; alpha <- 5 # get_interval <- function(){ cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) } } } gen_exp_var <- function() { data <- data.frame(x=rdply(trial_size, get_interval())/n) names(data) <- c("n", "x") return(data) } result <- rlply(alpha, gen_exp_var()) data <- data.frame(x=result[[1]]$x + result[[2]]$x + result[[3]]$x + result[[4]]$x + result[[5]]$x) # dens <- data.frame(y=dgamma(seq(0, 3,.01), shape=alpha, rate=np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Gamma")
41. 41. f(x, ↵, ) = ↵ (↵) x (↵+1) exp ✓ x ◆ (0  x < 1) Xi ⇠ Exp( ) Z = ↵X i=1 Xi ⇠ (↵, ) 1/Z ⇠ IG(↵, )
42. 42. trial_size = 7000; width <- .; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n; alpha <- 5 # get_interval <- function(){ cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) } } } gen_exp_var <- function() { data <- data.frame(x=rdply(trial_size, get_interval())/n) names(data) <- c("n", "x") return(data) } result <- rlply(alpha, gen_exp_var()) data <- data.frame(x=1/(result[[1]]$x + result[[2]]$x + result[[3]]$x + result[[4]]$x + result[[5]]$x)) # dens <- data.frame(y=dinvgamma(seq(0, 23,.01), shape=5, rate=1/np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Inversegamma")
43. 43. f(x) = ⇢ 1 (0  x  1) 0 (otherwise)
44. 44. Z = x1(1/2)1 + x2(1/2)2 + · · · + xq(1/2)q
45. 45. width <- 0.02 p <- 0.5; sample_size <- 1000 trial_size <- 100000 gen_unif_rand <- function() { # sample_size 2 # return (sum(rbern(sample_size, p) * (rep(1/2, sample_size) ** seq(sample_size)))) } gen_rand <- function(){ return( rdply(trial_size, gen_unif_rand()) ) } system.time(res <- gen_rand()) ggplot() + layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + ggtitle("Bernoulli to Standard Uniform")
46. 46. f(x, a, b) = ⇢ (b a) 1 (a  x  b) 0 (otherwise)
47. 47. a <- 5 b <- 8; width <- 0.05 p <- 0.5 sample_size <- 1000 trial_size <- 500000 gen_unif_rand <- function() { # sample_size 2 # return (sum(rbern(sample_size, p) * (rep(1/2, sample_size) ** seq(sample_size)))) } gen_rand <- function(){ return( rdply(trial_size, gen_unif_rand()) ) } system.time(res <- gen_rand()) res$V1 <- res$V1 * (b-a) + a ggplot() + layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + ggtitle("Bernoulli to Uniform") + xlim(4,9)
48. 48. f(x, ↵, ) = 1 B(↵, ) x↵ 1 (1 x) 1 (0 < x < 1) Xi ⇠ U(0, 1)iid (i = 1, 2, · · · , ↵ + 1)
49. 49. width <- 0.03; p <- 0.5 digits_length <- 30; set_size <- 3 trial_size <- 30000 gen_unif_rand <- function() { # digits_length 2 # return (sum(rbern(digits_length, p) * (rep(1/2, digits_length) ** seq(digits_length)))) } gen_rand <- function(){ return( rdply(set_size, gen_unif_rand())$V1 ) } unif_dataset <- rlply(trial_size, gen_rand, .progress='text') p <- ceiling(set_size * 0.5); q <- set_size - p + 1 get_nth_data <- function(a){ return(a[order(a)][p]) } disp_data <- data.frame(lapply(unif_dataset, get_nth_data)) names(disp_data) <- seq(length(disp_data)); disp_data <- data.frame(t(disp_data)) names(disp_data) <- "V1" x_range <- seq(0, 1, 0.001) dens <- data.frame(y=dbeta(x_range, p, q)*trial_size*width) ggplot() + layer(data=disp_data, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=x_range, y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Beta")
50. 50. E[X] = X( )P( ) + X( )P( ) = 0 ⇥ 0.8 + 1, 000, 000 ⇥ 0.2 = 200, 000 E[X] = X x xp(x) µ
51. 51. ✓ n x ◆ = n! (n x)!x! E[X] = nX x=0 xP(x) = nX x=0 x ✓ n x ◆ px (1 p)n x = nX x=0 x n! (n x)!x! px (1 p)n x = nX x=0 n (n 1)! (n x)!(x 1)! px (1 p)n x = np nX x=0 ✓ n 1 m 1 ◆ p(x 1) (1 p)(n 1) (x 1) = np = np nX x=1 ✓ n 1 m 1 ◆ p(x 1) (1 p)(n 1) (x 1) = np
52. 52. Var[X] = E[(X E[X])2 ] = X x (x E[x])2 P(x) = 2 µ
53. 53. Var[x] = E[(X E[X])2 ] = Z 1 1 (x E[x])2 f(x)dx = 2 E[X] = Z 1 1 xf(x)dx = µ
54. 54. E[g(X)] = Z 1 1 g(x)f(x)dx g(X) = (X E[X])2 E[ · ] = Z 1 1 · f(x)dx
55. 55. g(x) = xk E[g(X)] = E[Xk ] = Z 1 1 xk f(x)dx µ0 k
56. 56. g(x) = (x E[x])k E[g(X)] = E[(X E[X]])k ] = Z 1 1 (x E[x])k f(x)dx µk
57. 57. E[cX] = cE[X] * E[cX] = Z 1 1 cxf(x)dx = c Z 1 1 xf(x)dx = cE[X]
58. 58. Var[cX] = c2 Var[X] * Var[cX] = Z 1 1 (cx E[cx])2 f(x)dx = Z 1 1 (cx cµ)2 f(x)dx = Z 1 1 c2 (x µ)2 f(x)dx = c2 Z 1 1 (x µ)2 f(x)dx = c2 Var[X]
59. 59. P(x < X 5 x + x, y < Y 5 y + y) x, y ! 0 f(x, y) = lim x, y!0 P(x < X 5 x + x, y < Y 5 y + y) f(x, y)
60. 60. g(x) = Z 1 1 f(x, y)dy h(y) = Z 1 1 f(x, y)dx g(x) h(y)
61. 61. EX,Y [ g(X, Y )] = Z 1 1 Z 1 1 g(x, y)f(x, y)dxdy g(x, y) = x0.8 y0.8 (x, y) ⇠ N((4, 4), S) S =  1 0.5 0.4 1 EX,Y [ g(X, Y )] = 8.02
62. 62. g(X, Y ) = (X µX)(Y µY ) Cov[X, Y ] = E[(X µX)(Y µY )]
63. 63. g(X, Y ) = (X µX)(Y µY ) µX µX µX µX µY µY µY µY S1 = S2 = S3 = S4 =  1 0.8 0.8 1  1 0.8 0.8 1  1 0 0 1  1 0.999 0.999 1 Cov[X, Y ] = E[(X µX)(Y µY )] (x, y) ⇠ N((4, 4), S)
64. 64. f(x, y) f(x, y) = g(x)h(y)
65. 65. f(x, y) = g(x)h(y) = 0
66. 66. (x1, x2, · · · , xn) x1 f(x1) = Z · · · Z f(x1, · · · , xn)dx2 · · · dxn x1 f(x1, · · · , xn) = f(x1) · · · f(xn) x1 · · · xn
67. 67. x1 · · · xn g1(x1), · · · , gn(xn) x1 · · · xn E[ nY i=1 gi(xi)] = nY i=1 E[gi(xi)] E[g1(x1)] E[gn(xn)] E[ nY i=1 gi(xi)] = Z 1 1 · · · Z 1 1 g1(x1) · · · gn(xn)f(x1, · · · , xn)dx1 · · · dxn = Z 1 1 g1(x1)f(x1)dx1 · · · Z 1 1 gn(xn)f(xn)dxn = nY i=1 E[gi(xi)] f(x1) · · · f(xn)
68. 68. x1 · · · xn xi µi 2 i i = 1, 2, · · · , n c = (c1, · · · , cn) c1x1 + · · · + cnxn c1µ1 + · · · + cnµn c2 1 2 1 + · · · + c2 n 2 n
69. 69. E[c1x1 + · · · + cnxn] = Z 1 1 · · · Z 1 1 (c1x1 + · · · + cnxn)f(x1 · · · , xn)dx1 · · · dxn = c1 Z 1 1 · · · Z 1 1 x1f(x1 · · · , xn)dx1 · · · dxn · · · cn Z 1 1 · · · Z 1 1 xnf(x1 · · · , xn)dx1 · · · dxn =c1 Z 1 1 x1dx1 · · · cn Z 1 1 xndxn =c1µ1 + · · · + cnµn f(x1) · · · f(xn) f(x1) · · · f(xn) µ1 µn =c1 Z 1 1 x1dx1 · · · cn Z 1 1 xndxn =c1µ1 + · · · + cnµn
70. 70. Var[c1x1 + · · · + cnxn] = E[{(c1x1 + · · · + cnxn) E[c1x1 + · · · + cnxn]}2 ] = E[{c1(x1 µ1) + · · · + c1(x1 µ1)}2 ] = E[ nX i=1 c2 i (xi µi)2 + X i6=j cicj(xi µj)(xi µj)] = nX i=1 c2 i E[(xi µi)2 ] + X i6=j cicjE[(xi µj)(xi µj)] = c2 1 2 1 + · · · + c2 n 2 n c1µ1 + · · · + cnµn = E[xi µi]E[xj µj] = 0= 2 i
71. 71. x1 · · · xn x1 · · · xn xi µ 2 (µ, 2 )
72. 72. x1 · · · xn T = x1 + · · · + xn E[T] = E[x1 + · · · + xn] = E[x1] + · · · + E[xn] = nµ Var[T] = Var[x1 + · · · + xn] = Var[x1] + · · · + Var[xn] = n 2 2 1 = · · · = 2 n c1 = · · · = cn = 1 Var[c1x1 + · · · + cnxn] = c2 1 2 1 + · · · + c2 n 2 n
73. 73. ¯x = 1 n nX i=1 xi = 1 n T E[¯x] = 1 n E[T] = n · 1 n µ = µ Var[¯x] = Var[ 1 n T] = 1 n2 Var[T] = 2 n µ 2
74. 74. Var[¯x] = 2 n = 0.0833 500 = 0.000166 E[¯x] = 0.5
75. 75. Var[¯x]
76. 76. µ 2 P(|x µ| > ) 5 1 2 µ 2 1/ 2 = 1 ) P(|x µ| > ) 5 1 = 2 ) P(|x µ| > ) 5 1/4 = 3 ) P(|x µ| > ) 5 1/9
77. 77. 2 = Z 1 1 (x µ)2 f(x)dx = Z I1 (x µ)2 f(x)dx + Z I2 (x µ)2 f(x)dx + Z I3 (x µ)2 f(x)dx 2 = Z I1 (x µ)2 f(x)dx + Z I3 (x µ)2 f(x)dx = Z I1 2 2 f(x)dx + Z I3 2 2 f(x)dx = 2 2 [P(x 2 I1) + P(x 2 I3)] I1 = ( 1, µ ), I2 = [µ , µ + ], I3 = (µ + , 1) = P(|x µ| > ) P(|x µ| > ) 5 1 2 )
78. 78. x1 · · · xn µ 2 " > 0 lim n!1 P{|¯xn µ| = "} = 0 ¯xn = 1 n nX i=1 xi ¯xn µ ¯xn ! µ in P
79. 79. " > 0 P(|¯xn µ| > ") = P(|¯xn µ| > " p n p n ) 5 2 "2n = 2 ¯x= = 1 2
80. 80. f(x) = 1 p 2⇡ 2 exp ✓ (x µ)2 2 2 ◆ f(x) = 1 p 2⇡ exp ✓ x2 2 ◆ 1 < x < 1 1 < x < 1
81. 81. f(y) = y2
82. 82. f(x) = x2 f(y) = y2
83. 83. f(y) = exp( y2 )
84. 84. z = p 2y f(z) = exp ✓ 1 2 z2 ◆
85. 85. Z 1 1 e y2 dy = p ⇡ Z 1 1 exp ✓ z2 2 ◆ dz = p 2⇡ Z 1 1 1 p 2⇡ exp ✓ z2 2 ◆ dz = 1 dz = p 2dy
86. 86. Z 1 1 1 p 2⇡ exp ✓ z2 2 ◆ dz
87. 87. z = x µ dz dx = 1 f(x) = Z 1 1 1 p 2⇡ 2 exp ✓ (x µ)2 2 2 ◆ dx 1/
88. 88. D = (x1, · · · , xn) µ 2 ¯x µ / p n , n ! 1 N(0, 1) = 0.1, µ = 1 = 10, 2 = 1 2 = 100 ¯x = p n = r 1 2n = r 1 0.01 ⇥ 10000 = r 1 100 = 1 10
89. 89. g(x) = ext E[ext ] = Z 1 1 ext f(x)dx Mx(t) = E[ext ] Mx(t) My(t) x t = 0 y
90. 90. g(x) = ext ext = 1 + xt + t2 2! x2 + · · · + tk k! xk + · · · Mx(t) = E[ext ] = E[1 + xt + t2 2! x2 + · · · + tk k! xk + · · · ] = 1 + tE[x] + t2 2! E[x2 ] + · · · + tk k! E[xk ] + · · · = 1 + xµ0 1 + t2 2! µ0 2 + · · · + tk k! µ0 k + · · ·
91. 91. Mx(t) d dtk Mx(t) = E[xk ext ] t = 0 d dtk Mx(0) = E[xk ] = µ0 k
92. 92. x ⇠ N(µ, ) Mx(t) = E[ext ] = Z 1 1 ext 1 p 2⇡ 2 exp ✓ 1 2 (x µ)2 2 ◆ dx z = x µ x = µ + z dx = dz
93. 93. Mx(t) = Z 1 1 e(µ+ z)t 1 p 2⇡ 2 exp ✓ 1 2 z2 ◆ dz = eµt Z 1 1 1 p 2⇡ exp ✓ tz 1 2 z2 ◆ dz = eµt Z 1 1 1 p 2⇡ exp ✓ 1 2 [z2 2 tz 2 t2 + 2 t2 ] ◆ dz = eµt Z 1 1 1 p 2⇡ e 2t2 2 exp ✓ 1 2 (z t)2 ◆ dz = eµt e 2t2 2 Z 1 1 1 p 2⇡ exp ✓ 1 2 (z t)2 ◆ dz z t = w dz = dw Mx(t) = eµt e 2t2 2 Z 1 1 1 p 2⇡ exp ✓ w2 2 ◆ dw = eµt+ 2t2 2
94. 94. (f · g)0 = f0 · g + f · g0 (f g)0 (x) = f0 (g(x))g0 (x) M0 x(t) = (µ + 2 t)eµt+ 2t2 2 M00 x (t) = (µ + 2 t)2 ⇣ eµt+ 2t2 2 ⌘ + 2 ⇣ eµt+ 2t2 2 ⌘ = ⇣ eµt+ 2t2 2 ⌘ {(µ + 2 t)2 + 2 }
95. 95. Var[x] = E[x2 ] (E[x])2 = (µ2 + 2 ) (µ)2 = 2 Var[x] = E[(x E[x])2 ] = E[x2 2E[x]x + E[x]2 ) = E[x2 ] 2E[x]2 + E[x]2 = E[x2 ] E[x]2 t = 0 E[x] = M0 x(0) = (µ + 2 · 0)eµ·0+ 2·02 2 = µ E[x2 ] = M00 x (0) = ⇣ eµ·0+ 2·02 2 ⌘ {(µ + 2 · 0)2 + 2 } = µ2 + 2
96. 96. D = (x1, · · · , xn) µ 2 ¯x µ / p n , n ! 1 N(0, 1) T = x1 + · · · + xn T nµ p n 2T0 = T nµ p n = ¯x µ 1/ p n
97. 97. Mx(t) My(t) x t = 0 y T T0 = T nµ p n N(0, 2 )
98. 98. Mxi (t) = 1 + µ0 1t + µ0 2 t2 2! + µ0 3 t3 3! + · · · Mxi µ(t) = 1 + µ1t + µ2 t2 2! + µ3 t3 3! + · · · = 1 + 0 + 2 t2 2! + µ3 t3 3! + · · ·
99. 99. xi µ p n xi µ p n Mxi µ p n (t) = E[e xi µ p n t ] = 1 + 2 t2 2!n + µ3 t3 3!n3/2 + · · · + µk tk k!nk/2 + · · · = 1 + 2 t2 2n + n 2n = 1 2n n n ! 0 n ! 0 = 1 + 2 t2 + n 2n
100. 100. T0 = x1 µ p n + x2 nµ p n + · · · + xn µ p n = nX i=1 xi µ p n MT 0 (t) = MPn i=1 ⇣ xi µ p n ⌘(t) = E[e Pn i=1 ⇣ xi µ p n ⌘ t ] = nY i=0 E[e ⇣ xi µ p n ⌘ t ] = ✓ 1 + 1 n 2 t2 + n 2 ◆n er ⌘ lim n!1 ⇣ 1 + r n ⌘n r r = lim n!1 ⇣ 1 + r n ⌘n
101. 101. n ! 1 lim n!1 MT 0 = lim n!1 ✓ 1 + 1 n 2 t2 + n 2 ◆n = e 2t2 2 lim n!1 n = 0 N(0, 2 ) T0 = T nµ p n 2
102. 102. D = (x1, · · · , xn)
103. 103. ✓0 = ˆ✓(X1, · · · , Xn) ˆ✓lower(X1, · · · , Xn) 5 ✓0 5 ˆ✓upper(X1, · · · , Xn)
104. 104. ˆ✓(X)
105. 105. E[(ˆ✓(X) ✓)2 ]
106. 106. E[(ˆ✓(X) ✓)2 ] = E[{(E[ˆ✓(X)] ✓) + (ˆ✓(X) E[ˆ✓(X)])}2 ] = E[(E[ˆ✓(X)] ✓)2 + 2(E[ˆ✓(X)] ✓)(ˆ✓(X) E[ˆ✓(X)]) + (ˆ✓(X) E[ˆ✓(X)])2 ] = (E[ˆ✓(X)] ✓)2 + Var[ˆ✓(X)] E[ˆ✓(X)] ✓ E[(ˆ✓(X) ✓)2 ] = Var[ˆ✓(X)]
107. 107. E[¯x] = 1 n E[T] = n · 1 n µ = µ ¯x s2 = 1 n 1 nX i=1 (xi ¯x)2
108. 108. lim n!1 P{|¯xn µ| = "} = 0 ¯xn ! µ in P ˆ✓n(X) n ! 1 ˆ✓n(X) ! ✓ in P ˆ✓n(X) ¯xn µ
109. 109. Var[ˆ✓(X)] ˆ✓(X)
110. 110. D = (x1, · · · , xn) xi f(xi) nY i=1 f(xi) nY i=1 f(xi|✓) xi `(✓|x1, x2, · · · , xn) = nY i=1 f(xi|✓)
111. 111. x1, x2, · · · , x10 f(x1, x2, · · · , x10|µ, 2 ) = 10Y i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
112. 112. `(µ, 2 |x1, x2, · · · , x10) = 10Y i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
113. 113. ✓⇤ = arg max ✓ `(✓|x1, x2, · · · , xn) log `(✓|x1, · · · , xn) ⌘ L(✓|x1, · · · , xn) `
114. 114. µ, 2 L(µ, 2 |x1, x2, · · · , x10) = n 2 (2⇡) n 2 log 2 1 2 2 nX i=1 (xi µ)2 @L @µ = 1 2 2 nX i=1 (xi µ)2 ) nX i=1 xi = nµ ) µ⇤ = 1 n nX i=1 xi `(µ, 2 |x1, x2, · · · , xn) = nY i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
115. 115. @L @ 2 = n 2 1 2 + 1 2( 2)2 nX i=1 (xi µ)2 = 0 ) 1 2( 2)2 nX i=1 (xi µ)2 = n 2 2 ) 2⇤ = 1 n nX i=1 (xi µ)2 2⇤
116. 116. D = (x1, · · · , xn)µ 2 µ
117. 117. u ⇠ N(0, 1) t = u p v/m v ⇠ 2 (m) f(t) = m+1 2 p m⇡ m 2 ✓ t2 m + 1 ◆ m+1 2
118. 118. u ⇠ N(0, 1) v ⇠ 2 (m) v > 01 < u < +1 f(u, v) = 1 p 2⇡ exp ✓ u2 2 ◆ (1/2)n/2 (n/2) vn/2 1 e v/2 t = u p v/m x = v f(t) = m+1 2 p m⇡ m 2 ✓ t2 m + 1 ◆ m+1 2 (z) = Z 1 0 tz 1 e t dt
119. 119. µ D = (x1, · · · , xn) xi ⇠ N(µ, 2 ) ¯x ⇠ N(µ, 2 /n)¯x 1 2 nX i=1 (xi ¯x)2 ⇠ 2 n 1
120. 120. u = ¯x µ / p n ⇠ N(0, 1) v = 1 2 nX i=1 (xi ¯x)2 ⇠ 2 n 1 t = u p v/(n 1) = ¯x µ / p n · " 1 2 1 (n 1) nX i=1 (xi ¯x)2 # 1/2 = ¯x µ 1/ p n · 1 p s2 = ¯x µ s/ p n ⇠ tn 1 s2 = 1 n 1 nX i=1 (xi ¯x)2 s2
121. 121. P ✓ tn 1;↵/2 5 ¯x µ s/ p n 5 tn 1;↵/2 ◆ = 1 ↵ tn 1;↵/2 tn 1;↵/2 ↵/2 ↵/2 1 ↵ 1 ↵ 1 ↵ P ✓ ¯x tn 1;↵/2 s p n 5 µ 5 ¯x + tn 1;↵/2 s p n ◆ = 1 ↵ [ tn 1;↵/2, tn 1;↵/2] µ 1 ↵
122. 122. P ✓ tn 1;↵/2 5 ¯x µ s/ p n 5 tn 1;↵/2 ◆ = 1 ↵ tn 1;↵/2 tn 1;↵/2 ↵/2 ↵/2 1 ↵ 1 ↵ 1 ↵ P ✓ ¯x tn 1;↵/2 s p n 5 µ 5 ¯x + tn 1;↵/2 s p n ◆ = 1 ↵ [ tn 1;↵/2, tn 1;↵/2] µ 1 ↵
123. 123. = 1 µ = 0 H0 : µ0 = 0 H1 : µ 6= µ0
124. 124. ¯x = / p n / p 10 ; /3.16
125. 125. ↵/2 ↵/2 H0 : µ0 = 0
126. 126. H1 : µ = 1
127. 127. H1 : µ = 0.5
128. 128. H1 : µ = 3 µ0H1 : µ = 3 H0 : µ0 = 0
129. 129. e↵ect size : = µ µ0
130. 130. … … … … … … … …
131. 131. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
132. 132. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
133. 133. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
134. 134. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
135. 135. 1 n nX i=1 (xi ¯x)(yi ¯y)
136. 136. 1 n nX i=1 (xi ¯x)(yi ¯y)
137. 137. 1 n nX i=1 (xi ¯x)(yi ¯y)
138. 138. 1 n nX i=1 (xi ¯x)(yi ¯y)
139. 139. 1 n nX i=1 (xi ¯x)(yi ¯y)
140. 140. 1 n nX i=1 (xi ¯x)(yi ¯y)
141. 141. 1 n nX i=1 (xi ¯x)(yi ¯y)