By Prem C. Consul, Felix Famoye, Samuel Kotz
Lagrangian expansions can be utilized to procure various beneficial chance types, that have been utilized to actual lifestyles occasions together with, yet now not restricted to: branching tactics, queuing procedures, stochastic techniques, environmental toxicology, diffusion of data, ecology, moves in industries, revenues of recent items, and construction ambitions for max earnings. This publication offers a finished, systematic therapy of the category of Lagrangian likelihood distributions, besides a few of its households, their homes, and critical applications.
Key features:
* Fills a spot in ebook literature
* Examines many new Lagrangian chance distributions, their a variety of households, normal and particular homes, and functions to numerous various fields
* provides history mathematical and statistical formulation for simple reference
* certain bibliography and index
* workouts in lots of chapters
Graduate scholars and researchers with an exceptional wisdom of normal statistical concepts and an curiosity in Lagrangian likelihood distributions will locate this paintings helpful. 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 data, reliability, revenues administration, epidemiology, operations study, optimization in production and advertising, and infectious ailment keep an eye on will gain immensely from a number of the purposes within the book.
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Additional info for Lagrangian Probability Distributions
Example text
X n . Also, ⎤ ⎡ E⎣ s c j g j (X 1 , X 2 , . . , X n )⎦ = j=1 s c j E[g j (X 1 , X 2 , . . , X n )]. s are mutually independent, k k gi (X j ) = E i=1 E[gi (X j )]. v. X k , where k is an integer, is called the kth moment about zero and is denoted by µk = E[X k ]. 117) The first moment µ1 is also called the mean of the probability distribution and is denoted by µ. 118) which gives µ2 = µ2 − µ2 , µ3 = µ3 − 3µ2 µ + 2µ3 , µ4 = µ4 − 4µ3 µ + 6µ2 µ2 − 3µ4 , µ5 = µ5 − 5µ4 µ + 10µ3 µ2 − 10µ2 µ3 + 4µ5 .
It may be noted that the geometric distribution is a special case of the Consul distribution for m = 1; the Haight distribution is a special case of the Geeta distribution for m = 2; the Geeta distribution becomes degenerate when m → 1 as P(X = 1) → 1; when 0 < β = θ < 1 and b = (m − 1)θ, the Katz distribution reduces to the Geeta distribution and when β < 0 and β = −α, it becomes the Consul distribution by putting b = mα and α(1 + α)−1 = θ. 1 in the form (i) (ii) (iii) (iv) z = (u) = 1 − 1 − 4 pqu 2 /2qu , not the probability model as given in (7).
X n )⎦ = j=1 s c j E[g j (X 1 , X 2 , . . , X n )]. s are mutually independent, k k gi (X j ) = E i=1 E[gi (X j )]. v. X k , where k is an integer, is called the kth moment about zero and is denoted by µk = E[X k ]. 117) The first moment µ1 is also called the mean of the probability distribution and is denoted by µ. 118) which gives µ2 = µ2 − µ2 , µ3 = µ3 − 3µ2 µ + 2µ3 , µ4 = µ4 − 4µ3 µ + 6µ2 µ2 − 3µ4 , µ5 = µ5 − 5µ4 µ + 10µ3 µ2 − 10µ2 µ3 + 4µ5 . 3 Probabilistic and Statistical Results 17 The first central moment µ1 is always zero and the second central moment µ2 is called the variance of X and is denoted by V ar (X ) = σ 2 .