❚â♠ ❚➢t ❳→❝ ❙✉➜t ❱➔ ❚❤è♥❣ ❑➯ ❨ ❍å❝ ◆❣➔② ✷ t❤→♥❣ ✹ ♥➠♠ ✷✵✷✶ ❳→❝ ❙✉➜t ❈ì ❇↔♥ ❼ ❍♦→♥ ✈à✿ ❙è ❝→❝❤ s➢♣ ①➳♣ ♥❣➝✉ ♥❤✐➯♥ ❼ ❚ê ❤đ♣✿ ❙è ❝→❝❤ ❝❤å♥ ♥❣➝✉ ♥❤✐➯♥ ❧➦♣ ✈➔ ❦❤ỉ♥❣ ❝â ♣❤➙♥ ❜✐➺t t❤ù tü n ♣❤➛♥ tû Pn = n! k ♣❤➛♥ tû tø n n! Cnk = k!(n−k)! ♣❤➛♥ tû ✭k ≤ n✮ s❛♦ ❝❤♦ ❼ ❈❤➾♥❤ ❤ñ♣✿ ❙è ❝→❝❤ ❝❤å♥ ♥❣➝✉ ♥❤✐➯♥ k ♣❤➛♥ tû tø n! k ❦❤æ♥❣ ❧➦♣ ✈➔ ❝â ♣❤➙♥ ❜✐➺t t❤ù tü An = (n−k)! ❼ ❳❡♠ ❧↕✐ q✉② t➢❝ ❝ë♥❣ ✈➔ q✉② t➢❝ ♥❤➙♥ tr❛♥❣ ✶ tr♦♥❣ s→❝❤ ❣✐→♦ tr➻♥❤ ❼ ◗✉② t➢❝ ♣❤➛♥ ❜ò ①→❝ s✉➜t✿ ❼ ◗✉② t➢❝ ❝ë♥❣ ①→❝ s✉➜t✿ ❼ ◗✉② t➢❝ ❝ë♥❣ ①→❝ s✉➜t ❝❤♦ ✷ ❜✐➳♥ ❝è ✤ë❝ ❧➟♣✿ ❼ ❳→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥✿ ♥➳✉ ❼ ◗✉② t➢❝ ♥❤➙♥ ①→❝ s✉➜t✿ ❼ ◗✉② t➢❝ ♥❤➙♥ ①→❝ s✉➜t ❝❤♦ ❝→❝ ❜✐➳♥ ❝è ✤ë❝ ❧➟♣✿ ❼ ●✐↔ sû n ♣❤➛♥ tû ♣❤➛♥ tû ✤â ❦❤æ♥❣ s❛♦ ❝❤♦ k ♣❤➛♥ tû ✤â ¯ = − P (A) P (A) P (A + B) = P (A) + P (B) − P (AB) P (B) > t❤➻ P (A + B) = P (A) + P (B) P (A|B) = P (AB) P (B) P (AB) = P (A|B)P (B) A1 , A2 , · · · , An (n ≥ 2) P (AB) = P (A)P (B) ❧➔ ♠ët ♥❤â♠ ✤➛② ✤õ ❝→❝ sü ❦✐➺♥✳ ❳➨t sü ❦✐➺♥ ❇ s❛♦ ❝❤♦ ❇ ❝❤➾ ①↔② r❛ ❦❤✐ ♠ët tr♦♥❣ ❝→❝ sü ❦✐➺♥ A1 , A2 , · · · , An (n ≥ 2) s✉➜t ✤➛② ✤õ✳ ①↔② r❛✳ ❑❤✐ ✤â t❛ ❝â ❝æ♥❣ t❤ù❝ ①→❝ n P (B) = P (Ai )P (B|Ai ) i=1 ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ ❼ (k ≤ n) k ❇↔♥❣ ♣❤➙♥ ❜è ①→❝ s✉➜t✿ X=x P (X = x) x1 p1 x2 p2 ✶ ··· ··· xn pn ··· ··· ❼ pi = i ❼ P (a ≤ X ≤ b) = pi a≤xi ≤b ❼ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t✿ p1 p1 + p2 F (x) = p + p2 + · · · + pn−1 1 ❼ ❑ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✿ µ = E(X) = x < x1 , x1 ≤ x < x2 , x2 ≤ x < x , xn−1 ≤ x < xn , xn ≤ x xi p i i=1 ❼ ❼ Mode(X) ❼ X ✿ σ = Var(X) = E(X ) − [E(X)]2 P❤÷ì♥❣ s❛✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧➔ ❣✐→ trà ❝â ①→❝ s✉➜t ❧ỵ♥ ♥❤➜t✳ ❚r✉♥❣ ✈à ✭♠❡❞✐❛♥✮✿ ❚r✉♥❣ ✈à ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤✐➯♥ X X ỵ Med(X) tr ❝❤✐❛ ♣❤➙♥ ♣❤è✐ t❤➔♥❤ ✷ ♣❤➛♥ ❝â ①→❝ s✉➜t ❜➡♥❣ ♥❤❛✉✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧✐➯♥ tö❝ ❼ P (a < X < b) = b a fX (x)dx ❼ P (X < b) = P (−∞ < X < b) = b −∞ ❼ P (X > a) = P (a < X < +∞) = +∞ a ❼ fX (x)dx✳ fX (x)dx✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t x FX (x) = P (X ≤ x) = fX (t)dt −∞ ý ữỡ s + = E(X) = σ = Var(X) = E(X ) − [E(X)]2 xfX (x)dx; −∞ ❼ Mode(X) ❧➔ ❣✐→ trà ❧➔♠ ❝❤♦ ❤➔♠ ♠➟t ✤ë ✤↕t ❝ü❝ ✤↕✐✳ ✷ ▼ët sè ố tổ ỵ tự Pss X ∼ B(n, p) X ∼ Poisson(λ) X ∼ N(µ, σ ) ◆♦r♠❛❧ fX (x) Cnx px (1 − p)n−x λx −λ e x! (x−µ)2 − 2σ √ e 2πσ µ σ2 ♥♣ ♥♣✭✶✲♣✮ λ λ µ ỵ tt r x = n i=1 n xi = ❼ P❤÷ì♥❣ s❛✐ ♠➝✉ ❤✐➺✉ ❝❤➾♥❤ ❼ ✣ë ❧➺❝❤ ❝❤✉➞♥ ♠➝✉ ❤✐➺✉ ❝❤➾♥❤ ❼ ❚➛♥ s✉➜t ♠➝✉ f= k i=1 n s = s= n i=1 √ n i xi ( n xi )2 x2i − i=1 n n−1 = k i=1 ( ni x2i − n−1 k i=1 ni xi n ) s2 m(A) n ìợ ữủ ú t sû ❞ö♥❣ ❤➔♠ Φ(z) = ✶✳ ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ √1 2π z − t2 e dt −∞ µ ❼ ❚r÷í♥❣ ❤đ♣ ✤➣ ❜✐➳t ❼ ❚r÷í♥❣ ❤đ♣ ❝❤÷❛ ❜✐➳t σ ✱ n ≥ 30✿ x¯ − z α2 √sn ≤ µ ≤ x¯ + z α2 √sn ❼ ❚r÷í♥❣ ❤đ♣ ❝❤÷❛ ❜✐➳t σ ✱ n < 30✿ x¯ − t α2 ,n−1 √sn ≤ µ ≤ x¯ + t α2 ,n−1 √sn ❼ = z α2 √σn ❀ ❤♦➦❝ σ ✿ x¯ − z α2 √σn ≤ µ ≤ x¯ + z α2 √σn = z α2 √sn ❀ ❤♦➦❝ = t α2 ,n−1 √sn ✿ ữủ tữợ tố t ố ợ ữợ ữủ tr t✐♥ ❝➟② = z α2 σ( ❤♦➦❝ s) − α = 2Φ z α2 − ✷✳ ❑❤♦↔♥❣ t✐♥ ❝➟② ❝❤♦ ❼ nmin = z α2 · p✿ f − z α2 f (1−f ) n ≤ p ≤ f + z α2 f (1−f ) ✳ n f (1−f ) ✤÷đ❝ ❣å✐ ❧➔ ✤ë ❝❤➼♥❤ ①→❝ ❝❤♦ ÷ỵ❝ ❧÷đ♥❣✳ n ❼ ❑➼❝❤ t❤÷ỵ❝ ♠➝✉ tè✐ t❤✐➸✉ ✤è✐ ợ ữợ ữủ t t − α = 2Φ z α2 − ✸ nmin = f (1 − f ) zα/2 2 ❑✐➸♠ ✤à♥❤ ❣✐↔ t❤✉②➳t t❤è♥❣ ❦➯ ✶✳ ❑✐➸♠ ✤à♥❤ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❦❤✐ ❜✐➳t ❼ ▼æ ❤➻♥❤ ❦✐➸♠ ứ ự ỵ r tố ❼ ◆➳✉ |z| > z α2 α✱ z= H0 : µ = µ0 s✉② r❛ σ2 H1 : µ = µ0 ✈ỵ✐ z α2 ✳ √ (¯ x−µ0 ) n σ t❤➻ ❜→❝ ❜ä H0 ✱ ♥❣÷đ❝ ❧↕✐ t❤➻ ❝❤➜♣ ♥❤➟♥ ✷✳ ❑✐➸♠ ✤à♥❤ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝❤÷❛ t ổ ứ ự ỵ ♥❣❤➽❛ ❼ ❚rà t❤è♥❣ ❦➯ ❼ ◆➳✉ |z| > z α2 α✱ z= H0 : µ = µ0 s✉② r❛ n 30 H1 : = à0 ợ z α2 ✳ √ (¯ x−µ0 ) n s t❤➻ ❜→❝ ❜ä H0 ✱ ♥❣÷đ❝ ❧↕✐ t❤➻ ❝❤➜♣ ♥❤➟♥ ✸✳ ❑✐➸♠ ✤à♥❤ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝❤÷❛ ❜✐➳t ❼ ổ ứ ự ỵ ◆➳✉ |t| > t α2 ,n−1 σ2✱ H0 α✱ H0 : µ = µ0 σ2 ✈➔ H0 ✱ n < 30 H1 : = à0 ợ tr tt s✉② r❛ t❤➻ ❜→❝ ❜ä H0 t α2 ,n−1 ✳ ❚➼♥❤ ❣✐→ trà ❦✐➸♠ ✤à♥❤ ♥❣÷đ❝ ❧↕✐ t❤➻ ❝❤➜♣ ♥❤➟♥ H0 ✹✳ ❑✐➸♠ ✤à♥❤ ❣✐↔ t❤✉②➳t ✈➲ t✛ ❧➺ ❼ ổ ứ ự ỵ ❚➼♥❤ ❣✐→ trà ❦✐➸♠ ✤à♥❤ ❼ ❑➳t ❧✉➟♥✿ α✱ H0 : p = p0 s✉② r❛ zα/2 √ |z| > zα/2 H1 : p = p0 ✈ỵ✐ (f −p0 ) z=√ n p0 (1−p0 ) t❤➻ ❜→❝ ❜ä H0 ✱ ♥❣÷đ❝ ❧↕✐ t❤➻ ❝❤➜♣ ♥❤➟♥ ✹ H0 ✳ t= √ (¯ x−µ0 ) n s ❈→❝❤ t➻♠ ❼ zα ❦❤✐ trữợ ỷ tr tỵ✐ ❤↕♥ ❝❤✉➞♥ tr❛♥❣ ✶✵✹✲✶✵✺ s→❝❤ ❣✐→♦ tr➻♥❤ ✭❇↔♥❣ ♥➔② ❦❤ỉ♥❣ ✤÷đ❝ ♠❛♥❣ ✈➔♦ ♣❤á♥❣ t❤✐✮ ❼ ❈→❝❤ ✷✿ ❙û t s ữợ ổ ữợ ụ t ố ữợ số ữợ số ữợ r ữợ à=0 ✲❃ ❝❤♦ r❛ ♠ët ❝♦♥ sè ❝❤➼♥❤ ❧➔ 1−α ✲❃ ữợ =1 z s ữợ ổ ữợ số ữợ số ữợ ❆r❡❛ ❧➔ − α ✱ σ = 1✱ ✈➔ à=0 ổ ữợ r ởt ❝♦♥ sè ❝❤➼♥❤ ❧➔ zα ❱➼ ❞ư ✈ỵ✐ α = 0.025 t❤➻ z0.025 = 1.96 ✺