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Catégorie :Category: nCreator TI-Nspire
Auteur Author: usertest4804
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.24 Ko KB
Mis en ligne Uploaded: 18/04/2025 - 02:09:19
Uploadeur Uploader: usertest4804 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4587924
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.24 Ko KB
Mis en ligne Uploaded: 18/04/2025 - 02:09:19
Uploadeur Uploader: usertest4804 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4587924
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
<<
Binomial Distribution Use: Models number of successes in a fixed number of independent trials with same success probability. PMF: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k) Where: n = total number of trials k = number of successes p = probability of success (1 - p) = probability of failure Mean: ¼ = np Variance: ò = np(1 - p) Support: k = 0, 1, ..., n 2. Geometric Distribution Use: Models the number of trials until the first success. PMF: P(X = k) = (1 - p)^(k - 1) * p Where: k = trial on which first success occurs (k e 1) p = probability of success Mean: ¼ = 1 / p Variance: ò = (1 - p) / p² Memoryless Property: P(X > s + t | X > s) = P(X > t) 3. Negative Binomial Distribution Use: Number of trials needed to achieve r successes. PMF: P(X = k) = (k - 1 choose r - 1) * p^r * (1 - p)^(k - r) Where: r = number of desired successes k = total number of trials p = probability of success Mean: ¼ = r / p Variance: ò = r(1 - p) / p² 4. Hypergeometric Distribution Use: Number of successes in n draws without replacement. PMF: P(X = k) = [C(K, k) * C(N - K, n - k)] / C(N, n) Where: N = population size K = number of successes in population n = number of draws k = number of observed successes Mean: ¼ = n * (K / N) Variance: ò = n * (K / N) * (1 - K / N) * (N - n) / (N - 1) 5. Poisson Distribution Use: Counts events in a fixed interval (time, space, etc). PMF: P(X = k) = (e^(-») * »^k) / k! Where: » = expected number of events (rate) k = number of observed events Mean: ¼ = » Variance: ò = » 6. Normal Distribution PDF: f(x) = (1 / sqrt(2Àò)) * exp(- (x - ¼)² / (2ò)) Where: ¼ = mean à = standard deviation Standard Normal: Z = (X - ¼) / à 68-95-99.7 Rule: ~68% within 1Ã, ~95% within 2Ã, ~99.7% within 3à 7. Normal Approximation to Binomial Use: When np(1 - p) e 10 Approximation: X ~ Bin(n, p) H N(np, np(1 - p)) Continuity correction: P(X d k) H P(Y d k + 0.5) 8. Exponential Distribution Use: Models time between events in a Poisson process. PDF: f(x) = » * e^(-»x), for x e 0 Where: » = rate (events per unit time), » > 0 Mean: ¼ = 1 / » Variance: ò = 1 / »² Memoryless Property: P(X > s + t | X > s) = P(X > t) 9. Erlang Distribution Use: Models time until k events in a Poisson process. PDF: f(x) = (»^k * x^(k - 1) * e^(-»x)) / (k - 1)!, for x e 0 Where: k = number of events (integer) » = rate parameter Mean: ¼ = k / » Variance: ò = k / »² 10. Gamma Function and Gamma Distribution Gamma function: (n) = +^ x^(n - 1) * e^(-x) dx For integers: (n) = (n - 1)! Gamma PDF: f(x) = (1 / ((±) * ²^±)) * x^(± - 1) * e^(-x / ²), for x > 0 Where: ± = shape parameter ² = scale parameter Mean: ¼ = ±² Variance: ò = ±²² 11. Weibull Distribution Use: Life data analysis, reliability models. PDF: f(x) = (k / ») * (x / »)^(k - 1) * e^(-(x / »)^k), for x e 0 Where: k = shape parameter » = scale parameter Mean: ¼ = » * (1 + 1 / k) Variance: ò = »² * [(1 + 2 / k) - ((1 + 1 / k))²] 12. Lognormal Distribution Use: Models variables whose logarithm is normally distributed. If Y = ln(X) ~ N(¼, ò), then X ~ Lognormal(¼, ò) PDF: f(x) = (1 / (xÃ(2À))) * exp(- (ln(x) - ¼)² / (2ò)), for x > 0 Where: ¼ = mean of the logarithm à = std deviation of the logarithm Mean: E[X] = e^(¼ + ò / 2) Variance: Var(X) = (e^ò - 1) * e^(2¼ + ò) Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
Binomial Distribution Use: Models number of successes in a fixed number of independent trials with same success probability. PMF: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k) Where: n = total number of trials k = number of successes p = probability of success (1 - p) = probability of failure Mean: ¼ = np Variance: ò = np(1 - p) Support: k = 0, 1, ..., n 2. Geometric Distribution Use: Models the number of trials until the first success. PMF: P(X = k) = (1 - p)^(k - 1) * p Where: k = trial on which first success occurs (k e 1) p = probability of success Mean: ¼ = 1 / p Variance: ò = (1 - p) / p² Memoryless Property: P(X > s + t | X > s) = P(X > t) 3. Negative Binomial Distribution Use: Number of trials needed to achieve r successes. PMF: P(X = k) = (k - 1 choose r - 1) * p^r * (1 - p)^(k - r) Where: r = number of desired successes k = total number of trials p = probability of success Mean: ¼ = r / p Variance: ò = r(1 - p) / p² 4. Hypergeometric Distribution Use: Number of successes in n draws without replacement. PMF: P(X = k) = [C(K, k) * C(N - K, n - k)] / C(N, n) Where: N = population size K = number of successes in population n = number of draws k = number of observed successes Mean: ¼ = n * (K / N) Variance: ò = n * (K / N) * (1 - K / N) * (N - n) / (N - 1) 5. Poisson Distribution Use: Counts events in a fixed interval (time, space, etc). PMF: P(X = k) = (e^(-») * »^k) / k! Where: » = expected number of events (rate) k = number of observed events Mean: ¼ = » Variance: ò = » 6. Normal Distribution PDF: f(x) = (1 / sqrt(2Àò)) * exp(- (x - ¼)² / (2ò)) Where: ¼ = mean à = standard deviation Standard Normal: Z = (X - ¼) / à 68-95-99.7 Rule: ~68% within 1Ã, ~95% within 2Ã, ~99.7% within 3à 7. Normal Approximation to Binomial Use: When np(1 - p) e 10 Approximation: X ~ Bin(n, p) H N(np, np(1 - p)) Continuity correction: P(X d k) H P(Y d k + 0.5) 8. Exponential Distribution Use: Models time between events in a Poisson process. PDF: f(x) = » * e^(-»x), for x e 0 Where: » = rate (events per unit time), » > 0 Mean: ¼ = 1 / » Variance: ò = 1 / »² Memoryless Property: P(X > s + t | X > s) = P(X > t) 9. Erlang Distribution Use: Models time until k events in a Poisson process. PDF: f(x) = (»^k * x^(k - 1) * e^(-»x)) / (k - 1)!, for x e 0 Where: k = number of events (integer) » = rate parameter Mean: ¼ = k / » Variance: ò = k / »² 10. Gamma Function and Gamma Distribution Gamma function: (n) = +^ x^(n - 1) * e^(-x) dx For integers: (n) = (n - 1)! Gamma PDF: f(x) = (1 / ((±) * ²^±)) * x^(± - 1) * e^(-x / ²), for x > 0 Where: ± = shape parameter ² = scale parameter Mean: ¼ = ±² Variance: ò = ±²² 11. Weibull Distribution Use: Life data analysis, reliability models. PDF: f(x) = (k / ») * (x / »)^(k - 1) * e^(-(x / »)^k), for x e 0 Where: k = shape parameter » = scale parameter Mean: ¼ = » * (1 + 1 / k) Variance: ò = »² * [(1 + 2 / k) - ((1 + 1 / k))²] 12. Lognormal Distribution Use: Models variables whose logarithm is normally distributed. If Y = ln(X) ~ N(¼, ò), then X ~ Lognormal(¼, ò) PDF: f(x) = (1 / (xÃ(2À))) * exp(- (ln(x) - ¼)² / (2ò)), for x > 0 Where: ¼ = mean of the logarithm à = std deviation of the logarithm Mean: E[X] = e^(¼ + ò / 2) Variance: Var(X) = (e^ò - 1) * e^(2¼ + ò) Made with nCreator - tiplanet.org
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