第四讲 数学基础(III)：概率统计.pdf

1où êÆÄ:£III¤µVÇÚO •% E ŒÆ²LÆ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 1 / 29 ù̇SN 1. VÇØÄ: 1.1 Ÿo´VǺ 1.2 ‘ÅCþ†êiA 1.3 Œê½ÆÚ¥%4•½n 2. ÚOÆÄ: 2.1 Ÿo´ÚOƺ 2.2 †ÚOþ 2.3 b u 3. ²LA^ 3.1 ] ÂÃÇ ÚOA 3.2 Œ—^ž 1ATÍÕ1íº •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 2 / 29 l(½ ‘ŵ‰ÆgŽ ü? I VÇnØ ¤k@£ØdŠ3uµŒ5 ‘Åy– 8N1• )î‚ !š‘ Å5Æ" )) Gnedenko and Kolmogorov(1956) •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 3 / 29 l(½ ‘ŵ‰ÆgŽ ü? I VÇnØ ¤k@£ØdŠ3uµŒ5 ‘Åy– 8N1• )î‚ !š‘ Å5Æ" )) Gnedenko and Kolmogorov(1956) I Å ª‰»*@•§‰»XÓ˜‡ Œ ž¨Å짤k ÔNÑU옽 5 Æ$ħ¤kò5u) ¯‡Ñû½uL | ©ÙêÆ .5(½£ ^êÆ©Ù5£ã 0§½Â P (A|B) •¯‡ B u) e¯‡ A u) ^‡VÇ£conditional probability¤µ P (A|B) = •% (E ŒÆ²LÆ ) P (AB) P (B) êþ²LÆ£1où¤ 6 / 29 1. VÇØÄ: 1.1 Ÿo´VǺ ^‡Vdžn‡-‡úª (Ω, F , P ) •Vǘm§A, B ∈ F §… P (B) > 0§½Â P (A|B) •¯‡ B u) e¯‡ A u) ^‡VÇ£conditional probability¤µ P (A|B) = P (AB) P (B) I ¦{úªµP (AB) = P (B) P (A|B) •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 6 / 29 1. VÇØÄ: 1.1 Ÿo´VǺ ^‡Vdžn‡-‡úª (Ω, F , P ) •Vǘm§A, B ∈ F §… P (B) > 0§½Â P (A|B) •¯‡ B u) e¯‡ A u) ^‡VÇ£conditional probability¤µ P (A|B) = P (AB) P (B) I ¦{úªµP (AB) = P (B) P (A|B) I VÇúªµe {Bi , 1 ≤ i ≤ N } • P (A) = N X ˜m Ω ˜‡©)§Kµ P (Bi ) P (A|Bi ) i=1 •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 6 / 29 1. VÇØÄ: 1.1 Ÿo´VǺ ^‡Vdžn‡-‡úª (Ω, F , P ) •Vǘm§A, B ∈ F §… P (B) > 0§½Â P (A|B) •¯‡ B u) e¯‡ A u) ^‡VÇ£conditional probability¤µ P (A|B) = P (AB) P (B) I ¦{úªµP (AB) = P (B) P (A|B) I VÇúªµe {Bi , 1 ≤ i ≤ N } • P (A) = N X ˜m Ω ˜‡©)§Kµ P (Bi ) P (A|Bi ) i=1 I “d½n£Bayes’ theorem¤ µ P (A|B) = •% (E ŒÆ²LÆ ) P (A) P (B|A) P (B) êþ²LÆ£1où¤ 6 / 29 1. VÇØÄ: ~µÀ 7Þ n k˜U§˜ n 1.1 Ÿo´VǺ ¦²V< ¦²V<Ìĉ\u5˜°1’]Õ§ ß³ ,• ¦ò‡ŒÞ S4žE"˜±ƒ §ù n ¦² V< ýóA §ù| ¦ý Þ "1 Õ"ù˜g§ù ²V<@•,• ¦¬O§(Jù| ¦ý O " 10±L §ù° “ ±\q ˜Ï1’] 1’]ÕzÏÑk#ýÿ§ …¦‚ ÑA " •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 7 / 29 1. VÇØÄ: ~µÀ 7Þ n 1.1 Ÿo´VǺ ¦²V< k˜U§˜ n ¦²V<Ìĉ\u5˜°1’]Õ§ ß³ ,• ¦ò‡ŒÞ S4žE"˜±ƒ §ù n ¦² V< ýóA §ù| ¦ý Þ "1 Õ"ù˜g§ù ²V<@•,• ¦¬O§(Jù| ¦ý O " 10±L §ù° “ ±\q ˜Ï1’] 1’]ÕzÏÑk#ýÿ§ …¦‚ ÑA " 111±§\¬4ù n ¦²V<•\‰Ý]íº )) J. MÔË‚§5 •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ <êƵŒêâž“êÆg‘ åþ6 7 / 29 1. VÇØÄ: ~µ\ 1.1 Ÿo´VǺ Ø´™Ý©fíº bXk˜U§\uy Ø ¶iÑy3òÌuÙ /™Ý©fv¦ <0 ¶üþ§ …\ú òÌ 2·•rkXe©Ùµ Ñy3¶ü¥ vkÑy3¶ü¥ ´™Ý©f 10 9,990 Ø´™Ý©f 99,990 199,890,010 \ú \ Ø´™Ý©fíº )) J. MÔË‚§5 •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ <êƵŒêâž“êÆg‘ åþ6 8 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ‘ÅCþ (Ω, F , P ) •˜‡Vǘm§e½Â3 ˜mþ ¢Š¼ê X : Ω → R ÷v ∀x ∈ R kµ {ω : X (ω) ≤ x} ∈ F K¡ X • (Ω, F , P ) þ ‘ÅCþ" I lÑ.‘ÅCþµ ‘©Ù£Ëã|©Ù¤ !Ñt©Ù I ëY.‘ÅCþµþ!©Ù! ©Ù£pd©Ù¤ !•ê©Ù§±9t! F!χ ©Ù 2 •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 9 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ‘ÅCþ \È©ÙÚVǗݼê I X ´Vǘm (Ω, F , P ) þ ‘ÅCþ§eéu ∀x ∈ R •3µ F (x) = P (X ≤ x) ¡ F : R → [0, 1] •\È©Ù¼ê£cumulative distribution function§ cdf¤ " •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 10 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ‘ÅCþ \È©ÙÚVǗݼê I X ´Vǘm (Ω, F , P ) þ ‘ÅCþ§eéu ∀x ∈ R •3µ F (x) = P (X ≤ x) ¡ F : R → [0, 1] •\È©Ù¼ê£cumulative distribution function§ cdf¤ " I éëY‘ÅCþ X§XJØk•‡: dF f (x) = , dx •3µ Z x f (u) du = P (X ≤ x) = F (x) −∞ ¡ f : R → [0, ∞) •VǗݼê£probability density function§pdf¤ " •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 10 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ‘ÅCþ \È©ÙÚVǗݼê I X ´Vǘm (Ω, F , P ) þ ‘ÅCþ§eéu ∀x ∈ R •3µ F (x) = P (X ≤ x) ¡ F : R → [0, 1] •\È©Ù¼ê£cumulative distribution function§ cdf¤ " I éëY‘ÅCþ X§XJØk•‡: dF f (x) = , dx •3µ Z x f (u) du = P (X ≤ x) = F (x) −∞ ¡ f : R → [0, ∞) •VǗݼê£probability density function§pdf¤ " I éulÑ‘ÅCþ X ∈ {xi }N i=1 §¡ p (xi ) = P (X = xi ) •ªÇ¼ê £frequency function¤½©ÙÆ" •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 10 / 29 1. VÇØÄ: ~µ 1.2 ‘ÅCþ†êiA ©Ù x ∼ N (µ, σ 2 ) −(x−µ)2 1 1 √ e 2σ2 , F (x) = √ f (x) = σ 2π σ 2π ã¡5 µWikiz‰ •% ©Ù VÇ—Ý£†¤Ú\È©Ù£m¤¼ê" (E ŒÆ²LÆ ) êþ²LÆ£1où¤ Z x − e (t−µ)2 2σ 2 dt −∞ 11 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA êÆÏ"ÚÝ I éu R þ ¼ê g (x) Ú\È©Ù¼ê F (x) §½Â g (x) êÆÏ" £mathematical expectation¤•µ Z ∞ E [g (x)] = g (x)dF (x) −∞ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 12 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA êÆÏ"ÚÝ I éu R þ ¼ê g (x) Ú\È©Ù¼ê F (x) §½Â g (x) êÆÏ" £mathematical expectation¤•µ Z ∞ E [g (x)] = g (x)dF (x) −∞ I é?¿ k, l ∈ {1, 2...}§½Â‘ÅCþ Ý£moment¤µ •3§¡ƒ• X k :ݧþŠ E (x) •˜ :ݶ i k I e E (x − E (x)) •3§¡ƒ• X k ¥%ݧ• V ar (x) • I e E hx k
¥%ݶ h i I e E (x − E (x))k (y − E (y))l •3§¡ƒ• X Ú Y %ݧ •% • (E ŒÆ²LÆ ) Cov (x, y) ´ k+l ·Ü¥ ·Ü¥%Ý" êþ²LÆ£1où¤ 12 / 29 1. VÇØÄ: ~µ ©Ù N (µ, σ 2 ) I þŠ£mean¤ µµ§ I • I u¥ 1.2 ‘ÅCþ†êiA êiA êÚ¯ê £variance¤µσ 2 Ý£skewness¤µS = 0 I ¸Ý£kurtosis¤ µK = 3 ã¡5 µWikiz‰" •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 13 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ^‡êÆÏ"!‚5ÝK†‚5• I ‘ÅCþ X Ú Y §XJ R∞ ¦ O −∞ ydF (y|x) •3§K¡ƒ• Y 'u X = x ^‡êÆÏ" £conditional expectation¤ µ Z ∞ E (y|x) = Z ∞ ydF (y|x) = −∞ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ yf (y|x) dx −∞ 14 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ^‡êÆÏ"!‚5ÝK†‚5• I ‘ÅCþ X Ú Y §XJ R∞ ¦ O −∞ ydF (y|x) •3§K¡ƒ• Y 'u X = x ^‡êÆÏ" £conditional expectation¤ µ Z ∞ Z ∞ E (y|x) = ydF (y|x) = yf (y|x) dx −∞ I ‰½Ý X§·‚Žé ²¡ span (X) ¥Ú y l•á •þ ŷ§¦ ε = y − ŷ −∞ å •Ý kεk •á"· ‚¡ ŷ = Xβ • y 3 span (X) þ ‚5ÝK £projection¤"•Ò´`µ y = Xβ + ε ⇒ ŷ = E(y|X) = Xβ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 14 / 29 1. VÇØÄ: 1.2 ‘ÅCþ†êiA ^‡êÆÏ"!‚5ÝK†‚5• I ‘ÅCþ X Ú Y §XJ R∞ ¦ O −∞ ydF (y|x) •3§K¡ƒ• Y 'u X = x ^‡êÆÏ" £conditional expectation¤ µ Z ∞ Z ∞ E (y|x) = ydF (y|x) = yf (y|x) dx −∞ I ‰½Ý X§·‚Žé −∞ ²¡ span (X) ¥Ú y l•á •þ ŷ§¦ ε = y − ŷ å •Ý kεk •á"· ‚¡ ŷ = Xβ • y 3 span (X) þ ‚5ÝK £projection¤"•Ò´`µ y = Xβ + ε ⇒ ŷ = E(y|X) = Xβ I X T X Œ_ž§k‚5• ¦ O β̂ = arg min y − X β̂ = X T X
−1 XT y
β∈RK •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 14 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n Œê½Æ ïÄ3Ÿo^‡e§‘ÅCþ •% (E ŒÆ²LÆ ) þŠÂñ êþ²LÆ£1où¤ ˜‡~ê" 15 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n Œê½Æ ïÄ3Ÿo^‡e§‘ÅCþ I ½Â‘ÅS þŠÂñ ˜‡~ê" {ξn } •VÇÂñu a µ•3~ê a§∀ε > 0§k lim P (|ξn − a| < ε) = 1" n→∞ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 15 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n Œê½Æ ïÄ3Ÿo^‡e§‘ÅCþ I ½Â‘ÅS þŠÂñ ˜‡~ê" {ξn } •VÇÂñu a µ•3~ê a§∀ε > 0§k lim P (|ξn − a| < ε) = 1" n→∞ I ƒ Žnµ ‘ÅCþS {xi } Õá©Ù§…• k• 2 V ar (xi ) = σi < ∞ ž§c n ‘Žâ²þ x̄n •VÇÂñ ÙêÆÏ" þŠ µµ p x̄n → µ •% (E ŒÆ²LÆ ) Ù¥µx̄n = n n 1X 1X xi , µ = E (xi ) n i=1 n i=1 êþ²LÆ£1où¤ 15 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n ¥%4•½n ïÄ3Ÿo^‡e§‘ÅCþƒÚ ©ÙÂñ •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ ©Ù" 16 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n ¥%4•½n ïÄ3Ÿo^‡e§‘ÅCþƒÚ ©ÙÂñ I b½‘ÅCþS {xi } Õá©Ù§…• ©Ù" k• V ar (xi ) = σi2 < ∞ ž§½Â‘ ÅCþµ x̃n = •% (E ŒÆ²LÆ ) n 1 X Sn i=1 [xi − E (xi )], êþ²LÆ£1où¤ v u n uX Sn = t σi2 i=1 16 / 29 1. VÇØÄ: 1.3 Œê½ÆÚ¥%4•½n ¥%4•½n ïÄ3Ÿo^‡e§‘ÅCþƒÚ ©ÙÂñ I b½‘ÅCþS {xi } Õá©Ù§…• ©Ù" k• V ar (xi ) = σi2 < ∞ ž§½Â‘ ÅCþµ x̃n = I oäÊìŽnµ n 1 X Sn i=1 [xi − E (xi )], v u n uX Sn = t σi2 i=1 n → ∞ ž§þã‘ÅCþ x̃n ©ÙÂñ IO Z x 2 1 lim Fn (x) = lim P (x̃n ≤ x) = Φ (x) = e−t /2 dt n→∞ n→∞ 2π −∞ w P• Fn (x) → Φ (x)§dž¡ x̃n •©ÙÂñ£fÂñ¤uIO ©Ùµ ‘ÅCþ d x̃n → x" •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 16 / 29 2. ÚOÆÄ: 2.1 Ÿo´ÚOƺ ÚOÆ ÚOÆ£statistics¤´£ã˜X Œ^u£ã!)ºÚ©Û] ½ êâ óäÚEâ" I £ã£descriptive¤ÚOµ£ã £sample¤êâ A ¶ I íä£inferential¤ÚOµÄu êâíäoN£population¤ê â A " •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 17 / 29 2. ÚOÆÄ: †ÚOþ 2.2 †ÚOþ I {xi }N i=1 ´‘ÅCþ X N ‡ÕáÄ §†oN݃éA ÚOþ £statistic¤½ÂXeµ •% oNþŠ µ = E (x) oN• 2 σ oN k :Ý oN k ¥%Ý (E ŒÆ²LÆ ) þŠ = V ar (x)   E xk   E (x − µ)k • k :Ý k ¥%Ý êþ²LÆ£1où¤ 1 PN x̄ = N i=1 xi PN 2 1 s = N −1 i=1 (xi − x̄) PN k 1 Ak = N i=1 xi k 1 PN Mk = N i=1 (xi − x̄) 2 18 / 29 2. ÚOÆÄ: †ÚOþ 2.2 †ÚOþ I {xi }N i=1 ´‘ÅCþ X N ‡ÕáÄ §†oN݃éA ÚOþ £statistic¤½ÂXeµ oNþŠ µ = E (x) oN• 2 σ oN k :Ý oN k ¥%Ý þŠ = V ar (x)   E xk   E (x − µ)k 1 PN x̄ = N i=1 xi PN 2 1 s = N −1 i=1 (xi − x̄) PN k 1 Ak = N i=1 xi k 1 PN Mk = N i=1 (xi − x̄) 2 • k :Ý k ¥%Ý I 5¿µ I ÚOþ´ ¼ê§ ØÓÚOþ•ØÓ§ÏdÚOþ •´˜‡‘ ÅCþ¶ I ÚOþ Ñl •% ©Ù¡•Ä ©Ù£sample distribution¤§§†‘ÅCþ X ¤ oN©Ù£population distribution§F (x)¤´ü‡ØÓ Vg"' XŠâ¥%4•½n§ØØ X Ñl=«©Ù§3˜½^‡e§
ÑòìCÑl ©Ù N µ, σ 2 /N " þŠ x̄ (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 18 / 29 2. ÚOÆÄ: 2.2 †ÚOþ ~^ÚOþ I 8¥5•Iµ þŠ x̄ = xmean = E(x)!¥ ê xmed Ú¯ê xmode I CÉ5•Iµ 4 σext = max(x) − min(x)!• O σ 2 = V ar(x) = E(x − x̄)2 ÚI σ = sd(x) I ƒ'5•Iµ •% I • £covariance¤µCov(x, y) = E(x − x̄)E(y − ȳ) I Cov(x,y) ƒ'Xê£correlatin coefficient¤µCor(x, y) = sd(x)sd(y) (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 19 / 29 2. ÚOÆÄ: 2.2 †ÚOþ ƒ'Xê AÛ)º I •þ x, y ƒ'Xê£correlation coefficient¤†üöƒmY 'X•µ x·y ρ = Cor (x, y) = kxkkyk = cos (θ) üöÕáž ρ = 0 ⇔ x⊥y" •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 20 / 29 2. ÚOÆÄ: 2.2 †ÚOþ ƒ'Xê AÛ)º I •þ x, y ƒ'Xê£correlation coefficient¤†üöƒmY 'X•µ x·y ρ = Cor (x, y) = kxkkyk = cos (θ) üöÕáž ρ = 0 ⇔ x⊥y" I b½ x Ú y Ñl ©Ù§ ρ = 0 Ú ρ > 0 ž§kµ ã¡5 µhttp://www.cnblogs.com/vamei/p/3416138.html •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 20 / 29 2. ÚOÆÄ: ¢y©Ûl"b I "b ´ m© £Null hypothesis¤Ò´Ã ɽÃ'•ã§§Q´ïÄ å:• ½ ÄO§~Xµ I •% 2.3 b u AÚBvk O¶ I AéBvkK•¶ I =¦3^‡Ae§BéC•vkK•" (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 21 / 29 2. ÚOÆÄ: ¢y©Ûl"b I "b ´ 2.3 b u m© £Null hypothesis¤Ò´Ã ɽÃ'•ã§§Q´ïÄ å:• ½ ÄO§~Xµ I AÚBvk O¶ I AéBvkK•¶ I =¦3^‡Ae§BéC•vkK•" I "b Ò”/Âí½0§•kJø¿© KÒ•U yââUy²˜‡ 3§† ²Ñy ÞÚ O VÇŒu £negatively skewed¤ §`²½|eO'þ §ÝŒ" I \opd5(aggregational Gaussianity)µ • •% ©Ùƒ'¥yÑk¸þ— A §` ©Ù" I šé¡5(asymmetry)µ©Ù•3K Þ • ©Ù§XcÂÃdž (E ŒÆ²LÆ ) žmªÝþ,ž§ƒA ÂÃÇÚFÂÃǃ'§• êþ²LÆ£1où¤ Cu ÂÃǬª ©Ù" 27 / 29 3. ²LA^ 3.2 Œ—^ž 1ATÍÕ1íº Œ—^ž 1ATÍÕ1íº •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 28 / 29 3. ²LA^ 3.2 Œ—^ž 1ATÍÕ1íº Œ—^ž 1ATÍÕ1íº Source: Angrist and Pischke(2015). •% (E ŒÆ²LÆ ) êþ²LÆ£1où¤ 28 / 29 3. ²LA^ 3.2 Œ—^ž 1ATÍÕ1íº £OÍÏ J Yit = 167 − 29 Di − 49 P OSTt + 20.5 (Di × P OSTt ) + εit (8.8) •% (E ŒÆ²LÆ ) (7.6) (10.7) êþ²LÆ£1où¤ 29 / 29