Bài giảng Xác suất thống kê: Chương 4 - Tôn Thất Tú

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Bài giảng "Xác suất thống kê" Chương 4 - Thống kê mô tả, cung cấp cho sinh viên những kiến thức như: Tổng thể và mẫu; bảng tần số, tần suất; các số đặc trưng mẫu; biểu đồ; phân phối mẫu. Mời các bạn cùng tham khảo!

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Nội dung Text: Bài giảng Xác suất thống kê: Chương 4 - Tôn Thất Tú

❳⑩❈ ❙❯❻❚ ❚❍➮◆● ❑➊ ❚!♥ ❚❤➜% ❚& ✣➔ ◆➤♥❣✱ ✷✵✶✾ ❚!♥ ❚❤➜% ❚& ✶✴✷✻ ❈❤"#♥❣ ✹✿ ❚❤)♥❣ ❦➯ ♠- .↔ ✶✳ ❚-♥❣ .❤➸ ✈➔ ♠➝✉ ❛✳ ❈→❝ ❦❤→✐ ♥✐➺♠✿ ✲ ❚!♥❣ $❤➸ ✿ "➟♣ ❤&♣ "♦➔♥ ❜+ ❝→❝ ♣❤➛♥ "/ "❤0♥❣ ♥❤➜" "❤❡♦ ❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝9✉✳ ❚ "❤➸ ①❡♠ ♥❤K ♠+" ❜+ n ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {X1 , X2 , ..., Xn } ✤+❝ ❧➟♣ ✈➔ ❝> ❝[♥❣ ♣❤➙♥ ♣❤0✐ ✈O✐ X ✳ ❚!♥ ❚❤➜% ❚& ✷✴✷✻ ✲ ❈❤♦ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ {X1 , X2 , ..., Xn }✳ ❑❤✐ ✤>✿ ♠➝✉ $❤3❝ ♥❣❤✐➺♠✳ ✰ ❇+ n ❣✐→ "`a {x1 , x2 , ..., xn } ❝b "❤➸ c✉❛♥ "❤✉ ✤K&❝ ✤K&❝ ❣M✐ ❧➔ ✰ ▼+" ❤➔♠ ✭✤♦ ✤K&❝✮ T = T(X1 , X2 , ..., Xn ) ✤K&❝ ❣M✐ ❧➔ ♠+" $❤6♥❣ ❦➯ "`➯♥ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ {X1 , X2 , ..., Xn }✳ ❜✳ >❤➨♣ ❧➜② ♠➝✉ ✤E♥ ❣✐↔♥✿ ✲ ▲➜② ❝> ❤♦➔♥ ❧↕✐✿ ❈❤M♥ ♥❣➝✉ ♥❤✐➯♥ ♠+" ❝→ "❤➸ "Y "
✷✳ ❇↔♥❣ &➛♥ ()✱ &➛♥ (✉➜& ❛✳ ▼➝✉ ❦❤2♥❣ ❣❤➨♣ ❧6♣ ✲ ❇↔♥❣ %➛♥ '(✿ ●✐→ %-. x1 x2 ✳✳✳ xm ❚➛♥ '( n1 n2 ✳✳✳ nm %-♦♥❣ ✤3 x1 , ..., xm ❧➔ ❝→❝ ❣✐→ %-. ❦❤→❝ ♥❤❛✉ ✈
❱➼ ❞# ✸ ❈➙♥ #❤% ✶✵✵ #(→✐ #→♦ ✈-❛ #❤✉ ❤♦↕❝❤✱ #❛ ✤45❝ ❜↔♥❣ 9: ❧✐➺✉ 9❛✉✿ ❑❤:✐ ❧45♥❣ ✭❣✮ ❬✶✵✵❀✶✷✵✮ ❬✶✷✵❀✶✹✵✮ ❬✶✹✵❀✶✻✵✮ ❬✶✻✵❀✶✽✵✮ ❬✶✽✵❀✷✵✵✮ ❙: #(→✐ ✶✷ ✶✾ ✸✶ ✷✸ ✶✺ ❱➼ ❞# ✹ ❑❤↔♦ 9→# #❤K✐ ❣✐❛♥ #(✉♥❣ ❜➻♥❤ ✭#➼♥❤ ❜➡♥❣ ❣✐K✮ ♠➔ ♠Q# ♥❣4K✐ #- ✤Q #✉R✐ ✶✺ #(S ❧➯♥ ❞➔♥❤ ✤➸ ✤W❝ #✐♥ #X❝ #❤K✐ 9Y ♦♥❧✐♥❡ #(♦♥❣ ♠Q# ♥❣➔② S ♠Q# #❤➔♥❤ ♣❤:✱ 9: ❧✐➺✉ ✤45❝ #❤➸ ❤✐➺♥ S ❜↔♥❣ 9❛✉✿ ✣Q #✉R✐ ❬✶✺✱✷✵✮ ❬✷✵✱✸✵✮ ❬✸✵✱✹✵✮ ❬✹✵✱✺✵✮ ≥ 50 ❚❤K✐ ❣✐❛♥ #(✉♥❣ ❜➻♥❤ ✶✱✶ ✷✱✷ ✷✱✼ ✶✱✾ ✵✱✼ ❚!♥ ❚❤➜% ❚& ✼✴✷✻ ✸✳ ❈→❝ *+ ✤➦❝ ./0♥❣ ♠➝✉ ❛✳ ❚/✉♥❣ ❜➻♥❤ ✈➔ ♣❤0>♥❣ *❛✐ ♠➝✉ ❈❤♦ {x1 , x2 , ..., xn } ❧➔ ♠➝✉ () ❧✐➺✉ ❝-❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✳ ✲ ❚7✉♥❣ ❜➻♥❤ ♠➝✉✱ ❦➼ ❤✐➺✉ ❧➔ x✱ ✤=>❝ ?➼♥❤ ?❤❡♦ ❝A♥❣ ?❤B❝✿ n x1 + x2 + ... + xn 1X x= = xi n n i=1 ✲ D❤=E♥❣ (❛✐ ♠➝✉✱ ❦➼ ❤✐➺✉ ❧➔ s2 ✱ ✤=>❝ ?➼♥❤ ?❤❡♦ ❝A♥❣ ?❤B❝✿   n n 1 X 1  X s2 = (xi − x)2 = x2i − n(x)2  n−1 n−1 i=1 i=1 ✲ ✣G ❧➺❝❤ ❝❤✉➞♥ ♠➝✉✱ ❦➼ ❤✐➺✉ s✱ ✤=>❝ ?➼♥❤✿ √ s= s2 ❚!♥ ❚❤➜% ❚& ✽✴✷✻ ◆❤➟♥ ①➨.✿ ✐✮ ❑❤✐ ♠➝✉ ✤=>❝ ❝❤♦ K ❞↕♥❣ ❜↔♥❣ ?➛♥ ()✿ X x1 x2 ✳✳✳ xm ni n1 n2 ✳✳✳ nm ✲ ❑➼❝❤ ?❤=P❝ ♠➝✉✿ n = n1 + n2 + ... + nm ✳ 1 Pm ✲ ❚7✉♥❣ ❜➻♥❤ ♠➝✉✿ x = ni x i ✳ n i=1 " # 1 m ✲ D❤=E♥❣ (❛✐ ♠➝✉✿ s = 2 2 2 P ni xi − nx . n − 1 i=1 ❚!♥ ❚❤➜% ❚& ✾✴✷✻
✐✐✮ ❑❤✐ ♠➝✉ ' ❞↕♥❣ ❜↔♥❣ ❣❤➨♣ ❧1♣✿ X a0 − a1 a1 − a2 ✳✳✳ am−1 − am ni n1 n2 ✳✳✳ nm 45♦♥❣ ✤8ak−1 − ak = [ak−1 ; ak )✳ ak−1 + ak ✣➦4 xk = 4❛ ✤
❜✳ ❚#✉♥❣ ✈( ♠➝✉ ❙➢♣ ①➳♣ ♠➝✉ () ❧✐➺✉ -❤❡♦ -❤1 -2 -➠♥❣ ❞➛♥✱ ❣✐↔ (: x1 ≤ x2 ≤ ... ≤ xn ✳ !✉♥❣ ✈& ♠➝✉✱ ❦➼ ❤✐➺✉ xmed ✱ ①→❝ ✤A♥❤ ❜C✐✿  x n+1 ,  ♥➳✉ n ❧➫✱ xmed = x n2 + x n +1  2  2 ♥➳✉ n ❝❤➤♥✳ 2 ❝✳ ❍➺ ./ 012♥❣ 3✉❛♥ ♠➝✉ ❈❤♦ {(x1 , y1 ); (x2 , y2 ); ...; (xn , yn )} ❧➔ ♠➝✉ ❤❛✐ ❝❤✐➲✉ ❝K❛ ✈❡❝-M ♥❣➝✉ ♥❤✐➯♥ (X, Y )✳ ❍➺ +, -.♥❣ /✉❛♥ ♠➝✉ ✤OP❝ ①→❝ ✤A♥❤ ❜C✐✿ Pn i=1 (xi − x)(yi − y) r = pPn Pn 2 2 i=1 (xi − x) i=1 (yi − y) ❚!♥ ❚❤➜% ❚& ✶✸✴✷✻ ✸✳ ❇✐➸✉ ✤: ❛✳ ❇✐➸✉ ✤: ❝❤♦ ❞> ❧✐➺✉ #@✐ #↕❝✿ ✲ ❇✐➸✉ ✤5 ❝7 ✿ ✤R -❤A ❣R♠ ❝→❝ ❝S- ❤➻♥❤ ❝❤U ♥❤➟- ❝W ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ -➛♥ () ✭-➛♥ (✉➜-✮ -OM♥❣ 1♥❣✳ ✲ ✣❛ ❣✐→❝ ➛♥ +, ✿ ✤O\♥❣ ❣➜♣ ❦❤]❝ ♥)✐ ❝→❝ ✤✐➸♠ (x1 , n1 ), ..., (xk , nk ) -_➯♥ ♠➦- ♣❤➥♥❣✳ ✲ ✣❛ ❣✐→❝ ➛♥ +✉➜ ✿ ✤O\♥❣ ❣➜♣ ❦❤]❝ ♥)✐ ❝→❝ ✤✐➸♠ (x1 , f1 ), ..., (xk , fk ) -_➯♥ ♠➦- ♣❤➥♥❣✳ ✲ ❇✐➸✉ ✤5 ❤➻♥❤ !>♥✿ ❤➻♥❤ -_b♥ ✤OP❝ ❝❤✐❛ _❛ ❝→❝ ♣❤➛♥ ❝W ❞✐➺♥ -➼❝❤ -➾ ❧➺ ✈d✐ -➛♥ (✉➜- ✭-➛♥ ()✮ -OM♥❣ 1♥❣✳ ❚!♥ ❚❤➜% ❚& ✶✹✴✷✻ ❱➼ ❞E ✼ ❈❤♦ ❞B ❧✐➺✉✿ X ✶ ✷ ✸ ✹ ✺ ❚➛♥ +, ✶✷ ✸✵ ✷✵ ✶✶ ✼ ❚➛♥ +✉➜ ✵✳✶✺ ✵✳✹ ✵✳✸ ✵✳✶✹ ✵✳✶ ❍➣② ✈➩ ❝→❝ ❜✐➸✉ ✤5✿ ✲ ❜✐➸✉ ✤5 ❝7 ✭ ➛♥ +,✱ ➛♥ +✉➜ ✮ ✲ ✤❛ ❣✐→❝ ➛♥ +,✱ ➛♥ +✉➜ ✲ ❜✐➸✉ ✤5 ❤➻♥❤ !>♥ ❚!♥ ❚❤➜% ❚& ✶✺✴✷✻
❚!♥ ❚❤➜% ❚& ✶✻✴✷✻ ❚!♥ ❚❤➜% ❚& ✶✼✴✷✻ ❚!♥ ❚❤➜% ❚& ✶✽✴✷✻
❜✳ ❇✐➸✉ ✤' ❝❤♦ ❞, ❧✐➺✉ ❧✐➯♥ 12❝✿ ✲ ❚! ❝❤$❝ ✤& ✭❤✐)*♦❣-❛♠✮✿ ●✐↔ %& ♠➝✉ ❞+ ❧✐➺✉ ✤/0❝ ❝❤✐❛ ❧➔♠ m ❦❤♦↔♥❣ ✤➲✉ ♥❤❛✉ ✈;✐ ✤< ❞➔✐ ♠=✐ ❦❤♦↔♥❣ ❧➔ h✳ ❑❤✐ ✤@ AB ❝❤C❝ ✤D ❝E❛ ♠➝✉ ❞+ ❧✐➺✉ ♥➔② ❧➔ ♠
❇■➎❯ ✣➬ ❳⑩❈ ❙❯❻❚ ❈❍❯❽◆ 6.5 6.0 5.5 5.0 4.5 4.0 •1.5 •1.0 •0.5 0.0 0.5 1.0 1.5 ❚!♥ ❚❤➜% ❚& ✷✷✴✷✻ ✹✳ 1❤➙♥ ♣❤6✐ ♠➝✉ ✣;♥❤ ❧= ✶✿ ◆➳✉ {X1 , X2 , ..., Xn } ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝0 ♣❤➙♥ ♣❤3✐ ❝❤✉➞♥ N (µ; σ2 ) 5❤➻ √ σ2 (X − µ) n ❤❛② X ∼ N µ, ∼ N (0, 1). n σ ✣;♥❤ ❧= ✷✿ ◆➳✉ {X1 , X2 , ..., Xn } ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝0 ♣❤➙♥ ♣❤3✐ ❝❤✉➞♥ N (µ; σ2 ) 5❤➻✿ (n − 1)S 2 √ ∼ χ2n−1 , n(X − µ)/S ∼ Tn−1 σ2 ◆❤➟♥ ①➨D✿ ❚:; 30✮ 5❤➻ 5❤❡♦ ✤H♥❤ ❧I ❣✐B✐ ❤↕♥ 5:✉♥❣ 5➙♠✿ √ σ2 n(X − µ) X ❝0 ♣❤➙♥ ♣❤3✐ ①➜♣ ①➾ N µ, ✈➔ ❝0 ♣❤➙♥ ♣❤3✐ ①➜♣ ①➾ N (0, 1) n S ❚!♥ ❚❤➜% ❚& ✷✸✴✷✻ ❱➼ ❞H ✶✵ ❈❤✐➲✉ ❝❛♦ ❝(❛ )❤❛♥❤ ♥✐➯♥ )✉➙♥ )❤❡♦ ❧✉➟) ♣❤➙♥ ♣❤1✐ ❝❤✉➞♥ N (µ; σ2 )✱ ✈5✐ µ = 165✭❝♠✮ ✈➔ σ = 5✭❝♠✮✳ ❚➼♥❤ ①→❝ ?✉➜) ❝❤✐➲✉ ❝❛♦ )A✉♥❣ ❜➻♥❤ ❝(❛ ✶✻ )❤❛♥❤ ♥✐➯♥ ✤HI❝ ❝❤J♥ ♥❣➝✉ ♥❤✐➯♥ ❧5♥ ❤L♥ ✶✻✼ ✭❝♠✮✳ ●✐↔✐✳ ●P✐ Xi ❧➔ ❝❤✐➲✉ ❝❛♦ ❝-❛ 5❤❛♥❤ ♥✐➯♥ 5❤R i, i = 1, 16✳ ✣➦5 X̄ = (X1 + ... + X16 )/16✳ ❑❤✐ ✤0✱ X̄ ❝W♥❣ ❝0 ♣❤➙♥ ♣❤3✐ ❝❤✉➞♥ ✈B✐ 5:✉♥❣ ❜➻♥❤ µ = 165 ✈➔ ♣❤;X♥❣ Y❛✐ σ 2 /n = 25/16✳ ❳→❝ Y✉➜5 ❝➛♥ 5➻♠✿ 167 − 165 P (X̄ > 167) = 1 − P (X̄ ≤ 167) = 1 − Φ = 0.0548 5/4 ❚!♥ ❚❤➜% ❚& ✷✹✴✷✻
❱➼ ❞# ✶✶ ✣➸ ♥❣❤✐➯♥ ❝(✉ ✈➲ ,❤➙♠ ♥✐➯♥ ❝/♥❣ ,→❝ ✭,➼♥❤ ,34♥ ♥➠♠✮ ❝7❛ ♥❤➙♥ ✈✐➯♥ 9 ♠:, ❝/♥❣ ,② ❧=♥✱ ♥❣?@✐ ,❛ ❦❤↔♦ D→, ,❤➙♠ ♥✐➯♥ ❝7❛ ✶✵✵ ♥❤➙♥ ✈✐➯♥ ✤?H❝ ❝❤I♥ ♥❣➝✉ ♥❤✐➯♥ ,3♦♥❣ ❝/♥❣ ,②✳ ❑➳, N✉↔ ♥❤? D❛✉✿ ❚❤➙♠ ♥✐➯♥ ✺✲✼ ✽✲✶✵ ✶✶✲✶✸ ✶✹✲✶✻ ✶✼✲✶✾ ❙Z ♥❤➙♥ ✈✐➯♥ ✽ ✷✶ ✸✻ ✷✺ ✶✵ ❛✳ ❍➣② ,➼♥❤ ❣✐→ ,3^ ,3✉♥❣ ❜➻♥❤ ♠➝✉ ✈➔ ❣✐→ ,3^ ✤: ❧➺❝❤ ❝❤✉➞♥ ♠➝✉✳ ❜✳ ●✐↔ De ,❤➙♠ ♥✐➯♥ ❝/♥❣ ,→❝ ❝7❛ ♥❤➙♥ ✈✐➯♥ 9 ❝/♥❣ ,② ,3➯♥ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝f ❦g ✈I♥❣ ❧➔ ✶✷ ♥➠♠ ✈➔ ✤: ❧➺❝❤ ❝❤✉➞♥ ❧➔ ✸ ♥➠♠✳ ❚➼♥❤ ①→❝ D✉➜, ✤➸ ,3✉♥❣ ❜➻♥❤ ♠➝✉ ♥❤➟♥ ❣✐→ ,3^ ❧=♥ ❤k♥ ✶✷✱✺ ♥➠♠✳ ❚!♥ ❚❤➜% ❚& ✷✺✴✷✻ ●✐↔✐✳ ❉↕♥❣ $❤✉ ❣'♥✿ ❚❤➙♠ ♥✐➯♥ ✻ ✾ ✶✷ ✶✺ ✶✽ ❙5 ♥❤➙♥ ✈✐➯♥ ✽ ✷✶ ✸✻ ✷✺ ✶✵ ❛✳ ❈→❝ >5 ✤➦❝ $AB♥❣ ♠➝✉✿ x̄ = 12, 24; s = 3, 27✳ ❜✳ ❚❤❡♦ ✤G♥❤ ❧I ❣✐J✐ ❤↕♥ $A✉♥❣ $➙♠ X̄ ❝K ♣❤➙♥ ♣❤5✐ ①➜♣ ①➾ ❝❤✉➞♥ N (12; 32 /100) = N (12; 0, 09)✳ ❉♦ ✤K✿ 12, 5 − 12 P (X̄ > 12, 5) = 1 − Φ √ = 0.0478 0, 09 ❚!♥ ❚❤➜% ❚& ✷✻✴✷✻