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1 Mathematcal Statstcs / Chapter 6 radom varable probablty desty ucto. dstrbuto ucto.. jot desty ucto... k k margal desty ucto k m.. Statstcs ; θ ereces Y... p. ~ ; θ d radom sample... ; θ ~ statstc θ... samplg dstrbuto. MG uqueess.. θ µ.. T µ 0 s /... Y g : y Y y g... y y y. d Trasormato: Y y where Y g g Y dy MG : MG MG pd. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

2 Mathematcal Statstcs / The method o Dstrbuto uctos... Y h.... Y h... y. y Y y h.... y y y y y. EAMLE 6. 0 Y 3 E Y. Y rage [ ]. y + y y 3 Thereore y y y + y 9 EAMLE 6. U. 3 0 U 0 u. 3 u 3 uu0 u u Thereore 3 u u u 0 u 0 u ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

3 Mathematcal Statstcs / EAMLE 6.3 U U 0 u. u /0u u u / + u < u + u u0u u u< u EAMLE U. U 0 u. u U u u u u u u. u 0< µ. u [TI] u. u [ Y u u + Y u ] ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

4 Mathematcal Statstcs / EAMLE 6.5 Uorm0. g β g. 0. u U u g u g u g U ~ ep θ u t/ β u/ β u e dt e 0u. U 0. u / β e. U g β l. HOMEWORK #6- Let 0. d the pd o DUE 5 6 U HOMEWORK #6- Let ~ Normal0. Show that the pd o DUE 5 6 s χ HOMEWORK #6-3 DUE 5 6 Let e 0. d the pd o U ad the epected value o U. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

5 Mathematcal Statstcs / The method o Trasormato U h o-decreasg o-creasg. U h o-decreasg o-creasg. U h. h u d / du u. d U U u. du No-decreasg U u U u h u h u h u u u h Let h d u. The U u du No-creasg u U u U u h u h u h u u h Let h d u. The U u du EAMLE U 3. U u u+ u. 9 ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

6 Mathematcal Statstcs / 6. 9 EAMLE 6.7 Let 0. d the pd o U U 3 u U u u J JACOBIAN. U h V h h U V h U V UV u v h h J where J d d du dv d d du dv Jacoba. EAMLE e 0 < U +. U + V. 0 vu V U V. u u v e 0vu U u ue 0 u. UV U u ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

7 Mathematcal Statstcs / EAMLE U. U V. 0u v V U / V. UV u v v 0u v v u ulu0u. U EAMLE < < U + V. U + V. v U + V U V u v / u v B UV B u U u0< u u u< u < V v+ < v0 v v0< v< HOMEWORK #7- DUE 5 3 are a radom sample o sze rom e 0<. d the jot pd o. Suppose U + V. The d the jot pd o UV. u UV u v ep u v B d the margal pd o UV. The pd o U s a Gamma dstrbuto. The pd o V s a double epoetal dstrbuto. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

8 Mathematcal Statstcs / HOMEWORK #7- DUE 5 3 Let 0. U U HOMEWORK #7-6 DUE / /8 e 0 < U / 6.3 MG MG. U h. MG U MG. MG. EAMLE 6. ~ Normal µ σ. d the pd o Z µ σ stadardzato. µ < Z <. σ MG M µ t+σ t / t e. Z M Z σ t / tz t E e e. Z ~ Normal0 ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

9 Mathematcal Statstcs / EAMLE 6. Z ~ Normal0 U Z. 0 < U < tz M t E e E e U tu M U t t / U ~ Gamma α / β U χ λ. THEOREM... radom sample. MG M t M t M t... M t [ M t]. THEOREM... N µ σ. a. Normal µ a σ a. a a... a a a... a /. EAMLE 6. λ osso. Y e β 0 / β <. β λ..... U α β Gamma. β α β. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

10 Mathematcal Statstcs / U <. < M U tu t E e E e t t β U ~ Gamma β. HOMEWORK #7-3 DUE 5 3 Suppose Z Z... Z s a radom sample o sze rom N 0. Show that the pd o Z s χ λ. λ χ. HOMEWORK #7-4 DUE 5 3 Suppose ~ Normal0 σ ad are depedet. Show that U + ad V are depedet ad the margal pd o them s Normal 0σ respectvely. HOMEWORK #7-5 DUE 5 3 Let ~ Gamma α / β. Show U / β ~ χ λ. χ 6.4 Order Statstcs... radom sample Mmum value: m... Mamum value: ma... Rage: R Meda: + / / + / + ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

11 Mathematcal Statstcs / ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg EAMLE ] [... ] [ ] [... > > > ] [ EAMLE !...

12 Mathematcal Statstcs / EAMLE 6.5 /00 e 0 < The le-tme o the system s m. Sce /00 / 50 e e 0 <. E HOMEWORK #8- DUE 6 /00 e 0 < THEOREM k k! k k k k [ k ] [ k ] k < k < k! k! ROO k + k k. k + [ k rom < k ear k - k rom > k ] k k k + k k k k -k [ ] [ ] Q.E.D. ad j or < j! j k j j [ ] [ j ] [ k ] j! j! k! ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

13 Mathematcal Statstcs / EAMLE ad 4. Sce ! HOMEWORK #8- DUE 6 Suppose that... s a radom sample rom 0. d the pd o R. HOMEWORK #8-3 DUE 6 / β Suppose that... s a radom sample rom e 0 <. β ep β / 5 β / D o dscrete ad cotuous radom varables 6.5. Beroull Epermet. bary. p. ~ Beroull p p Beroull p pq ~ Bomal p p Bomal p pq ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

14 Mathematcal Statstcs / ~ Geometrc p p Geometrc / p q / p ~ NB r p r p Negatve Bomal r/ p rq/ p r r. ~ Hypergeometrc N M N M K Hyper-Geometrc M M N M N N N N N M N M M M N Bomal p. N ~ osso λ λ osso λ evet..... lm p q p λ λ λ e lm. EAMLE 6.7 Beroull p Bomal p. EAMLE 6.8 ~ Bomal p p ~ Bomal p + + p. EAMLE 6.9 p ~ Geometrc p p Negatve bomal NB p. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

15 Mathematcal Statstcs / 6. 0 EAMLE 6.0 ~ osso λ osso λ. EAMLE 6. ~ Geometrc p > a + b > a > b. Memoryless property Geometrc p Bomal p Beroull p λ p p M / N N Hypergeometrc N M NB r p r ossoλ 6.5. Relato betwee dscrete ad cotuous ~ Bomal p Normal p pq. W λ λw e w W w W > w 0 ~ osso λw e 0! w w λe λ ~ Epoetal β / λ. 0 λw λw Cotuous ~ Normal µ σ µ σ ormal ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

16 Mathematcal Statstcs / 6. 0 µ 0 σ stadard ormal. ~ Ep β β epoetal β λ λ β memoryless property. EAMLE 6. ~ Ep β > a + b > a > b. Memoryless property ~ Gamma α β α shape β scale gamma αβ αβ β α β. ~ Gamma α λ / β χ λ. EAMLE 6.3 ~ d Ep β. α αβ ~ Beta α β α β beta α + β α + β α + β + Gamma α β Beta α β. + r ~ t r r Studet t- 0 r W W ~ N 0 V ~ χ r ~ t r. V / r ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

17 Mathematcal Statstcs / ~ r r r r r - complcated r W / r W ~ χ r V ~ χ r ~ r r. V / r Cauchy Logormal e Normal 0 µ σ r r log µ +σ Bomal p oso λ p λ p µ pq σ Normal µ σ Ch squaredr Gamma α β α r / β Beta α β + α α /γ γ Epoetalβ r t r Webull γ EAMLE 6.4 ~ Gamma α β α β Gamma α β. EAMLE 6.5 ~ χ λ λ χ λ. ro. Sehyug Kwo Dept. o Statstcs HANNAM Uversty Sprg

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untitled Mathematics 4 Statistics / 6. 89 Chapter 6 ( ), ( /) (Euclid geometry ( ), (( + )* /).? Archimedes,... (standard normal distriution, Gaussian distriution) X (..) (a, ). = ep{ } π σ a 6. f ( F ( = F( f