Les statistiques en mathématiques (niveau Licence 2)

Résumé du document

Résumé des cours de statistiques de niveau Licence 2 présentant de nombreuses définitions, tableaux, calculs et explications qui permettent une parfaite compréhension du cours.

Extraits

[...] Moments centrés On appelle moment centré un moment pour a = picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 ì = f i i picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000) Cas particuliers : n = 0 ì 1 = n i i picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000) / n = n i x i / n - picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 n x 1 = ì 2 = f i x i - (picscalex1080100090000039b00000000003d000000000004000000030108000500000 00b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d0 0000000000c000d0000000000280000000d0000000c00000001000100000000000000000 00000000000000000000000000000000000000000ffffff003300c530120000000a00494 c0400ffff0400a702060011001d80000000000100000000003ff00000000000000000000 03d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c0 000000100010000000000000000000000000000000000000000000000000000000000fff fff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff 80101c0080001fff80101fff8ffff040000002701ffff030000000000)² = M 2 M 1 ì 3 = f i i picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000)³ = M 3 3 M 1 M 2 + 2 M 1 Ex Xn ix i - picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000(x i picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000)³03- 2.23 - 11.09 132- 1.23 - 1.86225 - 0.13 - B. Mesures de l'asymétrie Les caractéristiques de forme permettent de préciser l'allure de la courbe de fréquence sans la tracer. [...]

[...] Propriétés des indices de Laspeyres, Paasche et Fisher Circularité : Les 3 indices ne sont pas circulaires. L p t/0 L p t/t' * L p t'/0 * 1/100 Réversibilité : Les indices de Paasche et Laspeyres ne sont pas réversibles. L p t/0 / L p Mais L p 0/t = ( pⁿ 0 qⁿ / ( pⁿ t qⁿ = 1 / P p t/0 Cet inconvénient rend les indices de Laspeyres et Paasche inadaptés pour des comparaisons géographiques. [...]

[...] Ecart absolu moyen C'est la moyenne arithmétique des écarts par rapport à la valeur centrale (moyenne ou médiane). Ecart absolu moyen par rapport à picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 : ( Ó n i x i - picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 ) / n = Ó f i x i - picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 Ecart absolu moyen par rapport à Mé : ( Ó n i x i - Mé ) / n = Ó f i x i Mé Ex 15 : x = 30.75 Mé = 21 x i - Mé x i - picscalex1000100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 n i x i - Mé n i x i - picscalex1000100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 Ecart absolu moyen par rapport à picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 : 52875 / 4800 = 11 Ecart absolu moyen par rapport à Mé : 52720 / 4800 = 10.98 D. [...]

[...] t et 0 représentent soit des dates, soit des régions. Ex 22 : Un portefeuille d'actions est composé de plusieurs titres achetés sur plusieurs places financières (plusieurs monnaies), il a une valeur de ActionsEurosDollarsLivresq 0 (1999) 22531665.471104 q 1 (2000) 2673.51749 .06745EurosDollarsLivresp 0 (1999) 10.9511 .556p 1 (2000) Effet du taux de change dans l'évolution du portefeuille entre 1999 et 2000 : 2253 * 1 + 1665.47 * 1.151 + 1104 * 1.663 L p 2000/1999 = * 100 = * 1 + 1665.47 * 0.951 + 1104 * 1.556 Les variations du taux de charge ont pour effet d'augmenter la valeur du portefeuille de entre 1999 et 2000. [...]

[...] Moyenne arithmétique Elle est définie pour les variables quantitatives uniquement. Cas discret : picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 = (Ó n i x i ) / n = Ó f i x i Cas continu : picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 = Ó f i x i Avec x i = i i+1 le centre de i i+1 [ Ex 10 : picscalex1080100090000039b00000000003d0000000000040000000301080005000000 0b0200000000050000000c020d000e00030000001e003d000000410b8600ee000c000d00 000000000c000d0000000000280000000d0000000c000000010001000000000000000000 0000000000000000000000000000000000000000ffffff003300c530120000000a00494c 0400ffff0400a702060011001d80000000000100000000003ff000000000000000000000 3d000000410bc60088000c000d00000000000c000d0000000000280000000d0000000c00 00000100010000000000000000000000000000000000000000000000000000000000ffff ff00ccf80000edf80000f5f80101fbf8be30fbf80100f9f80001e2780101fff82600fff8 0101c0080001fff80101fff8ffff040000002701ffff030000000000 = 2510 E. [...]