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Catégorie :Category: nCreator TI-Nspire
Auteur Author: knfn
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 1.73 Ko KB
Mis en ligne Uploaded: 29/04/2025 - 08:04:00
Uploadeur Uploader: knfn (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4609662
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 1.73 Ko KB
Mis en ligne Uploaded: 29/04/2025 - 08:04:00
Uploadeur Uploader: knfn (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4609662
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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The measurement, x, is an unbiased estimate of u and has precision o. Expand in a Taylor series about the point x = u. Hence show that, if the coefficient of variation of x is small, the expected value of z = x^2 is given by u^2 + o^2 and that the coefficient of variation of z is approximately twice as large as that of x. Let x be an unbiased estimate of u with precision o (variance o^2). We define z = x^2. Perform a Taylor expansion of z around x = u: z = u^2 + 2u(x - u) + (x - u)^2 Take expectations: E[z] = u^2 + 2u*E[x - u] + E[(x - u)^2] = u^2 + 0 + o^2 = u^2 + o^2 Now compute Var(z): Using the delta method (first-order approximation): Var(z) = Var(2u*(x - u)) = 4u^2 * Var(x) = 4u^2 * o^2 Coefficient of variation for x: CV_x = sqrt(Var(x))/E[x] = o/u Coefficient of variation for z: CV_z = sqrt(Var(z))/E[z] = (2u*o)/(u^2 + o^2) For small CV_x (o/u << 1), u^2 + o^2 H u^2, so: CV_z = (2u*o)/u^2 = 2*(o/u) = 2*CV_x Thus: 1) E[z] = u^2 + o^2 2) CV_z = 2*CV_x when CV_x is small Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
The measurement, x, is an unbiased estimate of u and has precision o. Expand in a Taylor series about the point x = u. Hence show that, if the coefficient of variation of x is small, the expected value of z = x^2 is given by u^2 + o^2 and that the coefficient of variation of z is approximately twice as large as that of x. Let x be an unbiased estimate of u with precision o (variance o^2). We define z = x^2. Perform a Taylor expansion of z around x = u: z = u^2 + 2u(x - u) + (x - u)^2 Take expectations: E[z] = u^2 + 2u*E[x - u] + E[(x - u)^2] = u^2 + 0 + o^2 = u^2 + o^2 Now compute Var(z): Using the delta method (first-order approximation): Var(z) = Var(2u*(x - u)) = 4u^2 * Var(x) = 4u^2 * o^2 Coefficient of variation for x: CV_x = sqrt(Var(x))/E[x] = o/u Coefficient of variation for z: CV_z = sqrt(Var(z))/E[z] = (2u*o)/(u^2 + o^2) For small CV_x (o/u << 1), u^2 + o^2 H u^2, so: CV_z = (2u*o)/u^2 = 2*(o/u) = 2*CV_x Thus: 1) E[z] = u^2 + o^2 2) CV_z = 2*CV_x when CV_x is small Made with nCreator - tiplanet.org
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