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By Stephan Morgenthaler

Cet ouvrage divulge les principaux outils statistiques utiles à los angeles génétique: checks de signification, méthodes d'analyse fondées sur los angeles fonction de vraisemblance, algorithme EM, modélisations, examine de los angeles variance, classifications hiérarchiques, comparaisons multiples, and so forth. Tous permettent d'éclairer nombre de problèmes biologiques comme l. a. carcinogenèse, los angeles génétique des populations, los angeles sélection naturelle, les mutations, l'hérédité, les processus de coalescence, ou encore l'évolution. Ce livre s'adresse autant aux mathématiciens qu'aux biologistes. Rédigé avec un grand souci de clarté, il est également available aux non-spécialistes.

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S ♥♦tr❡ ❝❛s✱ ♦♥ ❛ E(C(1)) = φ′F (1) = 2b, E(C(2)) = φ′F (φF (1))φ′F (1) = (2b)2 ❡t ❡♥ ❣é♥ér❛❧ E(C(k)) = (2b)k ✳ ▲❛ t❛✐❧❧❡ ❡s♣éré❡ ❞✉ ❝❧♦♥❡ ❝r♦ît ❞♦♥❝ ❡①♣♦♥❡♥✲ t✐❡❧❧❡♠❡♥t ❡t ❞♦✉❜❧❡ ❡♥ ♠♦②❡♥♥❡ ❝❤❛q✉❡ (2b)k ≥ 2 ⇐⇒ k ≥ ln(2)/ ln(2b) ❣é♥ér❛t✐♦♥✳ ▲❡ t❛❜❧❡❛✉ ✷✳✷ ♠♦♥tr❡ q✉❡❧q✉❡s ❡①❡♠♣❧❡s✳ ✷✳✸✳✹ ❊①♣❛♥s✐♦♥ ❝❧♦♥❛❧❡ ❡♥ t❡♠♣s ❝♦♥t✐♥✉ ▲❡s ♣r♦❝❡ss✉s ❞❡ ❜r❛♥❝❤❡♠❡♥t ♠❛r❦♦✈✐❡♥ ❡♥ t❡♠♣s ❝♦♥t✐♥✉ ♣❡✉✈❡♥t êtr❡ ❞é❝r✐ts ♣❛r ❧❡s ♣r♦❜❛❜✐❧✐tés ❝♦♥❞✐t✐♦♥♥❡❧❧❡s s✉✐✈❛♥t❡s ✿ ⎧ ⎨ C(x) +1, C(x) −1, C(x + dx) = ⎩ C(x), ❛✈❡❝ ♣r♦❜❛❜✐❧✐té C(x)βdx + o(dx) ❛✈❡❝ ♣r♦❜❛❜✐❧✐té C(x)δdx + o(dx) s✐♥♦♥✳ ♦ù C(x) ❡st ❧❡ ♥♦♠❜r❡ ❞❡ ❝❡❧❧✉❧❡s ✜❧❧❡s q✉✐ ❡①✐st❡♥t ❡♥ t❡♠♣s x✳ ▲❡ ♣❛r❛♠ètr❡ β > 0 ❡st ✉♥ t❛✉① ❞❡ ❞✐✈✐s✐♦♥ ❡t δ > 0 ❡st ❧❡ t❛✉① ❞❡ ♠♦rt❛❧✐té✳ ❚♦✉t❡s ❧❡s ✷✽ ●é♥ét✐q✉❡ st❛t✐st✐q✉❡ ❚❛❜❧❡ ✷✳✷ ✕ ▲❡ t❛❜❧❡❛✉ ♠♦♥tr❡ ❧❛ ♣r♦❜❛❜✐❧✐té q✉✬✉♥❡ ❡①♣❛♥s✐♦♥ ❝❧♦♥❛❧❡ s✉r✈✐✈❡ ❡t ♥❡ s✬ét❡✐❣♥❡ ♣❛s✳ ▲❛ ❝♦❧♦♥♥❡ ❞❡ ❞r♦✐t❡ ❝♦♥t✐❡♥t ❧❡ ♥♦♠❜r❡ ❞❡ ❞✐✈✐s✐♦♥s ♥é❝❡ss❛✐r❡s ♣♦✉r ❞♦✉❜❧❡r ❧❛ t❛✐❧❧❡ ❞✉ ❝❧ô♥❡✳ b ✵✱✺✵✺ ✵✱✺✶✵ ✵✱✺✶✺ ✵✱✺✷✵ ✵✱✺✺✵ ✵✱✻✵✵ ♣r♦❜❛❜✐❧✐té ❞❡ s✉r✈✐❡ ❞✬✉♥❡ ❡①♣❛♥s✐♦♥ ❝❧♦♥❛❧❡ ✵✱✵✷ ✵✱✵✹ ✵✱✵✻ ✵✱✵✼✻ ✵✱✶✽ ✵✱✸✸ ♥♦♠❜r❡ ❞❡ ❣é♥ér❛t✐♦♥s ♣♦✉r ❞♦✉❜❧❡r ✼✵ ✸✺ ✷✹ ✶✽ ✽ ✹ ❝❡❧❧✉❧❡s ❛❣✐ss❡♥t ❞❡ ❧❛ ♠ê♠❡ ♠❛♥✐èr❡ ❡t ❛✉ t❡♠♣s x = 0 ❀ ✉♥❡ s❡✉❧❡ ❝❡❧❧✉❧❡ ✐♥✐t✐é❡ ❞é♠❛rr❡ ❧❡ ♣r♦❝❡ss✉s ❞❡ ❝r♦✐ss❛♥❝❡✳ ▲✬❡s♣ér❛♥❝❡ ❞❡ C(x) ✈ér✐✜❡ E(C(x + dx)) = E (C(x) + 1) C(x)βdx + (C(x) − 1) C(x)δdx + C(x)(1 − C(x)βdx − C(x)δdx) + o(dx) = E(C(x)) + (β − δ) E(C(x)) dx + 0(dx).

Dx ❊♥ ❞✐✈✐s❛♥t ❧✬✐♥t❡r✈❛❧❧❡ [0, x] ❡♥ [0, dx] ∪ (dx, x]✱ ♦♥ ♣❡✉t ❞é❞✉✐r❡ ✉♥❡ ❢♦r♠✉❧❡ ♣♦✉r ❧❛ ❝❤❛♥❝❡ ❞❡ s✉r✈✐❡ ❞❡ ❧✬❡①♣❛♥s✐♦♥ ❝❧♦♥❛❧❡✳ ❙♦✐t p(x) = P (C(x) = 0) ❧❛ ♣r♦❜❛❜✐❧✐té q✉❡ ❧❡ ❝❧♦♥❡ s✬❡st ét❡✐♥t à ❧✬â❣❡ x✳ ❖♥ ❛ p(x) = p(x − dx) [1 − (β + δ)dx] + βdxp(x − dx)2 + δ, ❝❛r ❞❛♥s ❧✬✐♥t❡r✈❛❧❧❡ [0, dx]✱ ❧❛ ❝❡❧❧✉❧❡ ✐♥✐t✐❛❧❡ ♣❡✉t s✉r✈✐✈r❡ s❛♥s ♠♦❞✐✜❝❛t✐♦♥✱ s❡ ♠✉❧t✐♣❧✐❡r ♣❛r ❞❡✉① ♦✉ s✬ét❡✐♥❞r❡✳ ❙✐ ❡❧❧❡ s✉r✈✐t✱ ❧❛ ❝❤❛♥❝❡ q✉❡ ❧❡ ❝❧ô♥❡ ❛✐❡ ❞✐s♣❛r✉ à ❧✬â❣❡ x ✈❛✉t p(x−dx)✱ s✐ ❛✉ ♠♦♠❡♥t dx ❧❡ ❝❧ô♥❡ ❝♦♥t✐❡♥t ❞❡✉① ❝❡❧❧✉❧❡s✱ ❧❛ ♣r♦❜❛❜✐❧✐té ❞❡✈✐❡♥t p(x−dx)2 ✱ ❝❛r ❧❡s ❞❡✉① ❝❡❧❧✉❧❡s ❛❣✐ss❡♥t ✐♥❞é♣❡♥❞❛♠♠❡♥t✳ ❙✐ ❧❛ ❝❡❧❧✉❧❡ ✐♥✐t✐❛❧❡ ♠❡✉rt✱ ❧❛ ♣r♦❜❛❜✐❧✐té q✉❡ ❧❡ ❝❧ô♥❡ ❛✐❡ ❞✐s♣❛r✉ à ❧✬â❣❡ x ✈❛✉t ✶✳ ▲✬éq✉❛t✐♦♥ ❝✐✲❞❡ss✉s ♠♦♥tr❡ q✉❡ p′ (x) = βp(x)2 − (β + δ)p(x) + δ.

Dx ❊♥ ❞✐✈✐s❛♥t ❧✬✐♥t❡r✈❛❧❧❡ [0, x] ❡♥ [0, dx] ∪ (dx, x]✱ ♦♥ ♣❡✉t ❞é❞✉✐r❡ ✉♥❡ ❢♦r♠✉❧❡ ♣♦✉r ❧❛ ❝❤❛♥❝❡ ❞❡ s✉r✈✐❡ ❞❡ ❧✬❡①♣❛♥s✐♦♥ ❝❧♦♥❛❧❡✳ ❙♦✐t p(x) = P (C(x) = 0) ❧❛ ♣r♦❜❛❜✐❧✐té q✉❡ ❧❡ ❝❧♦♥❡ s✬❡st ét❡✐♥t à ❧✬â❣❡ x✳ ❖♥ ❛ p(x) = p(x − dx) [1 − (β + δ)dx] + βdxp(x − dx)2 + δ, ❝❛r ❞❛♥s ❧✬✐♥t❡r✈❛❧❧❡ [0, dx]✱ ❧❛ ❝❡❧❧✉❧❡ ✐♥✐t✐❛❧❡ ♣❡✉t s✉r✈✐✈r❡ s❛♥s ♠♦❞✐✜❝❛t✐♦♥✱ s❡ ♠✉❧t✐♣❧✐❡r ♣❛r ❞❡✉① ♦✉ s✬ét❡✐♥❞r❡✳ ❙✐ ❡❧❧❡ s✉r✈✐t✱ ❧❛ ❝❤❛♥❝❡ q✉❡ ❧❡ ❝❧ô♥❡ ❛✐❡ ❞✐s♣❛r✉ à ❧✬â❣❡ x ✈❛✉t p(x−dx)✱ s✐ ❛✉ ♠♦♠❡♥t dx ❧❡ ❝❧ô♥❡ ❝♦♥t✐❡♥t ❞❡✉① ❝❡❧❧✉❧❡s✱ ❧❛ ♣r♦❜❛❜✐❧✐té ❞❡✈✐❡♥t p(x−dx)2 ✱ ❝❛r ❧❡s ❞❡✉① ❝❡❧❧✉❧❡s ❛❣✐ss❡♥t ✐♥❞é♣❡♥❞❛♠♠❡♥t✳ ❙✐ ❧❛ ❝❡❧❧✉❧❡ ✐♥✐t✐❛❧❡ ♠❡✉rt✱ ❧❛ ♣r♦❜❛❜✐❧✐té q✉❡ ❧❡ ❝❧ô♥❡ ❛✐❡ ❞✐s♣❛r✉ à ❧✬â❣❡ x ✈❛✉t ✶✳ ▲✬éq✉❛t✐♦♥ ❝✐✲❞❡ss✉s ♠♦♥tr❡ q✉❡ p′ (x) = βp(x)2 − (β + δ)p(x) + δ.

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