By Prem C. Consul, Felix Famoye, Samuel Kotz
ISBN-10: 0817643656
ISBN-13: 9780817643652
ISBN-10: 0817644776
ISBN-13: 9780817644772
Lagrangian expansions can be utilized to acquire a number of priceless likelihood versions, which were utilized to genuine lifestyles events together with, yet no longer constrained to: branching techniques, queuing procedures, stochastic approaches, environmental toxicology, diffusion of knowledge, ecology, moves in industries, revenues of latest items, and creation pursuits for maximum gains. This booklet offers a entire, systematic remedy of the category of Lagrangian chance distributions, in addition to a few of its households, their homes, and significant applications.
Key features:
* Fills a niche in publication literature
* Examines many new Lagrangian likelihood distributions, their various households, normal and particular houses, and purposes to quite a few varied fields
* provides history mathematical and statistical formulation for simple reference
* exact bibliography and index
* workouts in lots of chapters
Graduate scholars and researchers with an exceptional wisdom of ordinary statistical concepts and an curiosity in Lagrangian likelihood distributions will locate this paintings important. it can be used as a reference textual content or in classes and seminars on Distribution thought and Lagrangian Distributions. utilized scientists and researchers in environmental statistics, reliability, revenues administration, epidemiology, operations examine, optimization in production and advertising, and infectious sickness keep watch over will gain immensely from a few of the functions within the book.
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Extra info for Lagrangian Probability Distributions
Sample text
G j (t −1 ). Thus, if X 1 , X 2 , . . s and if their pgfs do exist, then their probability distributions do possess the convolution property. s X 1 and X 2 are closed under convolution. s possess this property. 6 Inference Let X 1 , X 2 , . . , X n be a random sample of size n taken from a population with cumulative distribution function F(X 1 , X 2 , . . , X n |θ1 , θ2 , . . , θk ), which depends upon k unknown parameters. s. X 1 , X 2 , . . , X n , say T j ≡ T j (X 1 , X 2 , . . , X n ), j = 1, 2, 3, .
6. The means and variances of some Lagrangian distributions L 1 ( f 1 ; g; y) No. Distribution 1. Hari 2. Weighted delta-binomial 3. Weighted delta Poisson 4. Weighted delta- Mean µ Variance σ 2 1+2q−4q 2 (1−2q)2 n + mpq 1−mp (1−mp)2 n + λ 1−λ (1−λ)2 nq mp q−mp + (q−mp)2 2 12q + 4q (1+2q) (1−2q)3 (1−2q)4 2 (n+1)mpq + 2m(m−1) p4 q (1−mp)3 (1−mp) 2 (n+1)λ + 2λ 4 (1−λ)3 (1−λ) 2 (n+1)mpq + 2m(m+1) p4 q (q−mp)3 (q−mp) np mpq 1−mp + (1−mp)2 2 (n+mp) pq + 2m(m−1) p4 q (1−mp)3 (1−mp) np λ 1−λ + (1−λ)2 np1 mpq 1−mp + (1−mp)2 θ λ 1−λ + (1−λ)2 mpq θ 1−mp + (1−mp)2 θq mp q−mp + (q−mp)2 2 npq+nλp 2 +λ + 2λ 4 (1−λ)3 (1−λ) np1 q1 +mpq+mnpp1 (q−q1 ) (1−mp)3 2 + 2m(m−1) p4 q (1−mp) θ+λ + 2λ2 (1−λ)3 (1−λ)4 2 θ+mpq−θmp 2 + 2m(m−1) p4 q (1−mp)3 (1−mp) 2 θq 3 +mpq(1+θ p) + 2m(m+1) p4 q (q−mp)3 (q−mp) np λ q(1−λ) + (1−λ)2 4 λq 2 +np−λnp 2 +2(1+λ)λ2q 2 + 2λ 4 q 2 (1−λ)3 (1−λ) negative binomial 5.
Poisson-binomial eθ(z−1) 16. )−1 p × 2 F0 (−y, −my; ; θq ) qm (1− pz)m negative binomial 17. Negative binomial- qn (1− pz)n eλ(z−1) q1n (1− p1 z)n qm (1− pz)m 19. Negative binomial-binomial 20. Logarithmic-binomial q1n (1− p1 z)n y my−1 (q − mp)e−θ θ qy! ×2 F0 (−y, my; ; −θp ) (1 − λ)q n e−yλ (yλ) y! (q − mp)q1n p1 q my−1 (n+y−2)! y! (n) ×2 F1 (−y, my; 1 − y − n; p/ p1 ) y my y ×2 F1 (n, −y; my − y + 1; p−1pq ) (q + pz)m (1 − mp)q1n q my ( p/q) y ln(1+ pz/q) (q + pz)m p y q my−y (1−mp) (− ln q) (− ln q) × (−1)k 21.
Lagrangian Probability Distributions by Prem C. Consul, Felix Famoye, Samuel Kotz
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