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tr t trr t t Prt r trs rtt tts

  1. ■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑

  2. ❆♥♥♦✉♥❝❡♠❡♥ts ◮ ❆ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ✐s ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ◮ ❊①❛♠ ✷ ✐s ♥❡①t ❲❡❞♥❡s❞❛②✳

  3. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡♠✐♥❞❡rs ❯s❡ ❝♦♥t✐♥✉♦✉s ♣r♦❜❛❜✐❧✐t② ✇❤❡♥ ♣✐❝❦✐♥❣ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡rs✳ ◮ ❙❛♠♣❧❡ s♣❛❝❡s ❛♥❞ ❡✈❡♥ts ❛r❡ ♠❛❞❡ ✉♣ ♦❢ ✐♥t❡r✈❛❧s✳ ◮ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♠✐♥✉s t❤❡ ❧❡❢t ❡♥❞♣♦✐♥t✳ ◮ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❡✈❡♥t E ✐s t❤❡ ❧❡♥❣t❤ ♦❢ E ❞✐✈✐❞❡❞ ❜② t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳ ◮ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ✐♥t❡r✈❛❧s ✐s t❤❡ ✐♥t❡r✈❛❧ ❢♦r♠❡❞ ❜② t❤❡✐r ♦✈❡r❧❛♣✳

  4. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡♠✐♥❞❡rs ❯s❡ ❝♦♥t✐♥✉♦✉s ♣r♦❜❛❜✐❧✐t② ✇❤❡♥ ♣✐❝❦✐♥❣ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡rs✳ ◮ ❙❛♠♣❧❡ s♣❛❝❡s ❛♥❞ ❡✈❡♥ts ❛r❡ ♠❛❞❡ ✉♣ ♦❢ ✐♥t❡r✈❛❧s✳ ◮ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♠✐♥✉s t❤❡ ❧❡❢t ❡♥❞♣♦✐♥t✳ ◮ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❡✈❡♥t E ✐s t❤❡ ❧❡♥❣t❤ ♦❢ E ❞✐✈✐❞❡❞ ❜② t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳ ◮ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ✐♥t❡r✈❛❧s ✐s t❤❡ ✐♥t❡r✈❛❧ ❢♦r♠❡❞ ❜② t❤❡✐r ♦✈❡r❧❛♣✳

  5. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✼ − ✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✻ − ✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  6. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✼ − ✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✻ − ✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  7. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✶✼ − ✶✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✶✻ − ✶✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  8. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✶✼ − ✶✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✶✻ − ✶✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  9. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✶✼ − ✶✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✶✻ − ✶✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  10. ❈♦♥t✐♥✉♦✉s Pr♦❜❛❜✐❧✐t② ❘❡✈✐❡✇ ❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ s♣❛❝❡ Ω = [ ✶✵ , ✶✼ ] ❛♥❞ ❡✈❡♥t ✭✐♥t❡r✈❛❧✮ F = [ ✶✸ , ✶✻ ] ✿ ◮ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ❤❛s ❧❡♥❣t❤ ✶✼ − ✶✵ = ✼ . ◮ ❊✈❡♥t F ❤❛s ❧❡♥❣t❤ ✶✻ − ✶✸ = ✸ . ◮ ❍❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ✐s ▲❡♥❣t❤ ♦❢ F ❚♦t❛❧ ❧❡♥❣t❤ = ✸ ✼ . ◆♦t✐❝❡ t❤❛t F t❛❦❡s ✉♣ ✸✴✼t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳

  11. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω ✳ ❚❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡✈❡♥t F ❣✐✈❡♥ t❤❛t E ♦❝❝✉rr❡❞ ✐s P ( F | E ) = ▲❡♥❣t❤ ♦❢ E � F . ▲❡♥❣t❤ ♦❢ E

  12. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ✭❉❡t❛✐❧s✮ ▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω ✳ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t P ( F | E ) = P ( E � F ) . P ( E ) ■♥ t❡r♠s ♦❢ ❧❡♥❣t❤s✱ ✇❡ ❤❛✈❡ ▲❡♥❣t❤ ♦❢ E � F ❚♦t❛❧ ❧❡♥❣t❤ P ( F | E ) = , ▲❡♥❣t❤ ♦❢ E ❚♦t❛❧ ❧❡♥❣t❤ ❛♥❞ t❤✐s s✐♠♣❧✐✜❡s t♦ t❤❡ ❢r❛❝t✐♦♥ P ( F | E ) = ▲❡♥❣t❤ ♦❢ E � F ▲❡♥❣t❤ ♦❢ E .

  13. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ✶ ▲❡t Ω = [ ✶✵ , ✶✼ ] ✱ E = [ ✶✶ , ✶✻ ] ✱ ❛♥❞ F = [ ✶✷ , ✶✺ ] ✳ ▲❡t ✉s ❝♦♠♣✉t❡ P ( F | E ) ✳ ◮ ❋✐♥❞ E � F ❛♥❞ ✐ts ❧❡♥❣t❤✿

  14. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ✶ ▲❡t Ω = [ ✶✵ , ✶✼ ] ✱ E = [ ✶✶ , ✶✻ ] ✱ ❛♥❞ F = [ ✶✷ , ✶✺ ] ✳ ▲❡t ✉s ❝♦♠♣✉t❡ P ( F | E ) ✳ ◮ ❋✐♥❞ E � F ❛♥❞ ✐ts ❧❡♥❣t❤✿

  15. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ✶ ▲❡t Ω = [ ✶✵ , ✶✼ ] ✱ E = [ ✶✶ , ✶✻ ] ✱ ❛♥❞ F = [ ✶✷ , ✶✺ ] ✳ ▲❡t ✉s ❝♦♠♣✉t❡ P ( F | E ) ✳ ◮ ❋✐♥❞ E � F ❛♥❞ ✐ts ❧❡♥❣t❤✿

  16. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❢♦r ■♥t❡r✈❛❧s ✶ ▲❡t Ω = [ ✶✵ , ✶✼ ] ✱ E = [ ✶✶ , ✶✻ ] ✱ ❛♥❞ F = [ ✶✷ , ✶✺ ] ✳ ▲❡t ✉s ❝♦♠♣✉t❡ P ( F | E ) ✳ ◮ ❋✐♥❞ E � F ❛♥❞ ✐ts ❧❡♥❣t❤✿ E ❛♥❞ F ♦✈❡r❧❛♣ ♦♥ [ ✶✷ , ✶✺ ] , ✇❤✐❝❤ ❤❛s ❧❡♥❣t❤ ✶✺ − ✶✷ = ✸✳ ◮ ▲❡♥❣t❤ ♦❢ E ✐s ✶✻ − ✶✶ = ✺✳ ◮ ❍❡♥❝❡ P ( F | E ) = ▲❡♥❣t❤ ♦❢ E � F = ✸ ✺ . ▲❡♥❣t❤ ♦❢ E ◆♦t✐❝❡ t❤❛t E � F t❛❦❡s ✉♣ ✸✴✺t❤s ♦❢ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ E ✳

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