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❲❤❛t ✐s t❤❡ ✧❍♦t ❍❛♥❞✧❄ ❆ ❇❛②❡s✐❛♥ ❚❡st ❢♦r t❤❡ ✧❍♦t ❍❛♥❞✧ ❉❛r❥❛ ❚✉ts❝❤❦♦✇ ✶ ✱ ❈♦♥♦r ❉♦❧❛♥ ✷ ✱ ●✐❧❧❡s ❉✉t✐❧❤ ✷ ✱ ❘✉✉❞ ❲❡t③❡❧s ✷ ✱ ❙♦♣❤✐❡ ✈♦♥ ❞❡r ❙❧✉✐s ✸ ✱ ❊r✐❝✲❏❛♥ ❲❛❣❡♥♠❛❦❡rs ✷ • ✒❚❤❡ ✱❤♦t ❤❛♥❞✬ ❛♥❞ ✱str❡❛❦ s❤♦♦t✐♥❣✬✲t❡r♠s r❡❢❡r t♦ t❤❡ ❜❡❧✐❡❢ ❉❛r❥❛ ❚✉ts❝❤❦♦✇ ✶ t❤❛t t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ ♣❧❛②❡r ❞✉r✐♥❣ ❛ ♣❛rt✐❝✉❧❛r ♣❡r✐♦❞ ✐s s✐❣♥✐✜❝❛♥t❧② ❜❡tt❡r t❤❛♥ ❝♦✉❧❞ ❜❡ ❡①♣❡❝t❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ❞❛r❥❛✳t✉ts❝❤❦♦✇❅st✉❞❡♥t✳✉♥✐✲t✉❡❜✐♥❣❡♥✳❞❡ ♣❧❛②❡r❵s ♦✈❡r❛❧❧ r❡❝♦r❞✳✏ ✭●✐❧♦✈✐❝❤✱ ❱❛❧❧♦♥❡ ❛♥❞ ❚✈❡rs❦②✱ ✶✾✽✺✮ • ❚❤❡ s❛♠❡ ❝❧❛✐♠ ✇❛s ♠❛❞❡ ❜② ●✐❧❞❡♥ ✫ ❲✐❧s♦♥ ✭✶✾✾✺✱ ✶ ❊❜❡r❤❛r❞ ❑❛r❧s ❯♥✐✈❡rs✐tät ❚ü❜✐♥❣❡♥ ❈♦❣♥✐t✐✈❡ Ps②❝❤♦❧♦❣②✮ ❛❜♦✉t ♣❡♦♣❧❡✬s ♣❡r❢♦r♠❛♥❝❡ ✐♥ s✐♠♣❧❡ ✷ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r❞❛♠ ♣❡r❝❡♣t✉❛❧ t❛s❦s✳ ✸ ❋r❡❡ ❯♥✐✈❡rs✐t②✱ ❆♠st❡r❞❛♠ ✷✴✾✴✷✵✶✷ ✷✵✵✺ ❇❛tt✐♥❣ ❖✉t❝♦♠❡s ❢r♦♠ ❈❛r❧♦s ●✉✐❧❧❡♥ ❖✉t❧✐♥❡ ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✶ , ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✶ , ✶ , ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✶ , ✵ , ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✶ , ✶ , ✵ , ✵ , • ❈✉rr❡♥t t❡sts ❢♦r str❡❛❦✐♥❡ss ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✵ , ✶ , ✶ , ✶ , ✶ , ✵ , ✵ , ✶ , ✶ , ✶ , ✶ , ✵ , ✵ , ✶ , ✵ , ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✵ , ✶ , ✵ , • ❆ ❇❛②❡s✐❛♥ t❡st ❢♦r str❡❛❦✐♥❡ss ✵ , ✵ , ✵ , ✶ , ✶ , ✶ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✶ , ✵ , • ❆♣♣❧✐❝❛t✐♦♥ t♦ r❡❛❧ ❞❛t❛ ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✶ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , • ❊❛s② t♦ ❞❡t❡❝t str❡❛❦✐♥❡ss❄ ✶ , ✵ , ✵ , ✶ , ✶ , ✶ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✶ , ✶ , ✵ , ✶ , ✶ , ✶ , ✵ , ✶ , ✵ , ✵ , ✵ , ✵ , ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✵ , ✵ , ✶ , ✵ , ✵ , ✵ , ✶ ■s ❈❛r❧♦s ●✉✐❧❧❡♥ ❛ str❡❛❦② ♣❧❛②❡r❄

❖✉t❧✐♥❡ ❈✉rr❡♥t ❚❡sts ❢♦r ❙tr❡❛❦✐♥❡ss ❊①❛♠♣❧❡s • ❈✉rr❡♥t t❡sts ❢♦r str❡❛❦✐♥❡ss • ❚❤❡ ❧♦♥❣❡st r✉♥ ♦❢ ❤✐ts ✭❆❧❜❡rt✱ ✷✵✵✽✮ • ❆ ❇❛②❡s✐❛♥ t❡st ❢♦r str❡❛❦✐♥❡ss • ❘✉♥s t❡st ✭●✐❧❧♦✈✐❝❤✱ ❱❛❧♦♥❡ ✫ ❚✈❡rs❦②✱ ✶✾✽✺✮ • ❆♣♣❧✐❝❛t✐♦♥ t♦ r❡❛❧ ❞❛t❛ • ❚❡st ♦❢ st❛t✐♦♥❛r✐t② ✭●✐❧❧♦✈✐❝❤✱ ❱❛❧♦♥❡ ✫ ❚✈❡rs❦②✱ ✶✾✽✺✮ • ❊❛s② t♦ ❞❡t❡❝t str❡❛❦✐♥❡ss❄ • ❇❧❛❝❦ st❛t✐st✐❝ ✭❆❧❜❡rt✱ ✷✵✵✽✮ ❈✉rr❡♥t ❚❡sts ❢♦r ❙tr❡❛❦✐♥❡ss ❖✉t❧✐♥❡ Pr♦❜❧❡♠s • ❊①✐st✐♥❣ t❡sts ❛r❡ ♠♦st❧② ❝❧❛ss✐❝❛❧ ♦r ❢r❡q✉❡♥t✐st✱ ❛♥❞ ♦♥❧② • ❈✉rr❡♥t t❡sts ❢♦r str❡❛❦✐♥❡ss ❝♦♥s✐❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ • ❆ ❇❛②❡s✐❛♥ t❡st ❢♦r str❡❛❦✐♥❡ss • ❚❤❡ t❡sts ❤❛✈❡ ✈❡r② ❧♦✇ ♣♦✇❡r✳ • ❆♣♣❧✐❝❛t✐♦♥ t♦ r❡❛❧ ❞❛t❛ • ❚❤✐s ♠❡❛♥s t❤❛t ✐t ✐s ♥♦t ✈❡r② ✐♥❢♦r♠❛t✐✈❡ ✇❤❡♥ ♦♥❡ ✏❢❛✐❧s t♦ • ❊❛s② t♦ ❞❡t❡❝t str❡❛❦✐♥❡ss❄ r❡❥❡❝t t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s✑✳ • ❚❤❡ t❡sts s♦♠❡t✐♠❡s ✉s❡ ❛❞✲❤♦❝ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ❞❛t❛ ✐♥ ❡♣♦❝❤s✳ ❇✉t t❤❡ s✐③❡ ♦❢ t❤❡ ❡♣♦❝❤ ❛✛❡❝ts t❤❡ r❡s✉❧t ✭❜❧❛❝❦ st❛t✐st✐❝✮✳

❆ ❇❛②❡s✐❛♥ ❚❡st ❢♦r ❙tr❡❛❦✐♥❡ss ❆ ❇❛②❡s✐❛♥ ❚❡st ❢♦r ❙tr❡❛❦✐♥❡ss ❚❤❡ ❍▼▼ • ❆ss✉♠❡ ❛ st❛t❡ s♣❛❝❡ { ❙ t : t ∈ N } ✇✐t❤ t✇♦ ♣♦ss✐❜❧❡ st❛t❡s • ❲❡ ✇❛♥t t♦ ❛ss❡ss t❤❡ ❡✈✐❞❡♥❝❡ ❢♦r ❛♥❞ ❛❣❛✐♥st t❤❡ ❤②♣♦t❤❡s✐s ❙ t ∈ { ✵ , ✶ } ✿ ♦❢ str❡❛❦② ♣❡r❢♦r♠❛♥❝❡✳ • ❲❡ ❝♦♥tr❛st t✇♦ ♠♦❞❡❧s✿ • ❚❤❡ ❝♦♥st❛♥t✲♣❡r❢♦r♠❛♥❝❡ ♠♦❞❡❧✭❈♣▼✮ • ❆ t❤r❡❡✲♣❛r❛♠❡t❡r ❤✐❞❞❡♥ ▼❛r❦♦✈ ♠♦❞❡❧✭❍▼▼✮ ✕ t❤✐s ♠♦❞❡❧ ✐s ✐♥ ❧✐♥❡ ✇✐t❤ ♦♥❡✬s ✐♥t✉✐t✐♦♥ ♦❢ str❡❛❦✐♥❡ss✳ • ❚❤❡ st❛t❡ s♣❛❝❡ { ❙ t : t ∈ N } s❛t✐s✜❡s t❤❡ ▼❛r❦♦✈ ♣r♦♣❡rt②✿ Pr ( ❙ t = s t | ❙ ( t − ✶ ) = s ( t − ✶ ) , . . . , ❙ ✶ = s ✶ ) = Pr ( ❙ t = s t | ❙ ( t − ✶ ) = s ( t − ✶ ) ) ❆ ❇❛②❡s✐❛♥ ❚❡st ❢♦r ❙tr❡❛❦✐♥❡ss ❆ ❇❛②❡s✐❛♥ ❚❡st ❢♦r ❙tr❡❛❦✐♥❡ss ❚❤❡ ❍▼▼ ❇❛②❡s ❋❛❝t♦r • ❆❢t❡r s❡❡✐♥❣ t❤❡ ❞❛t❛✱ ✇❤✐❝❤ ♠♦❞❡❧ ✐s ♣r❡❢❡r❛❜❧❡❄ • ❚❤❡ ♦♥❡ ✇✐t❤ t❤❡ ❤✐❣❤❡r ♣♦st❡r✐♦r ♣r♦❜❛❜✐❧✐t②✦ Pr ( ❍▼▼ | ❉❛t❛ ) Pr ( ❈♣▼ | ❉❛t❛ ) = Pr ( ❉❛t❛ | ❍▼▼ ) Pr ( ❉❛t❛ | ❈♣▼ ) • ❇❡❢♦r❡ s❡❡✐♥❣ t❤❡ ❞❛t❛ ❜♦t❤ ♠♦❞❡❧s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡q✉❛❧❧② ❧✐❦❡❧② ⇒ Pr ( ❍▼▼ ) Pr ( ❈♣▼ ) = ✶ ✇❤❡r❡ • ❚♦ ❝❤♦♦s❡ ❛ ♠♦❞❡❧ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❇❛②❡s ❢❛❝t♦r ✭❇❋✮ • ♣ ❤ = Pr ( ✶ | ❙ t = ✶ ) ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❤✐t ✐♥ t❤❡ ❤♦t st❛t❡ Pr ( ❉❛t❛ | ❍▼▼ ) • ♣ ❝ = Pr ( ✶ | ❙ t = ✵ ) ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❤✐t ✐♥ t❤❡ ❝♦❧❞ st❛t❡ Pr ( ❉❛t❛ | ❈♣▼ ) • ❚❤❡ ❇❛②❡s ❢❛❝t♦r ✐s t❤❡ ❝❤❛♥❣❡ ❢r♦♠ ♣r✐♦r t♦ ♣♦st❡r✐♦r ♦❞❞s • α = Pr ( ❙ t = ✶ | ❙ ( t − ✶ ) = ✵ ) = Pr ( ❙ t = ✵ | ❙ ( t − ✶ ) = ✶ ) ✐s t❤❡ ❜r♦✉❣❤t ❛❜♦✉t ❜② t❤❡ ❞❛t❛✳ ♣r♦❜❛❜✐❧✐t② ♦❢ s✇✐t❝❤✐❣ ❜❡t✇❡❡♥ st❛t❡s • ◗✉❛♥t✐✜❡s t❤❡ ❡✈✐❞❡♥❝❡ ❢♦r ♦♥❡ ✈❡rs✉s t❤❡ ♦t❤❡r ♠♦❞❡❧ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❞❛t❛✳