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The Algebra of Random Variables

The Algebra of Random Variables,M. D. Springer

The Algebra of Random Variables   (Citations: 205)
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Published in 1979.
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    • ...We refer to 16 or 15 for its definition, properties and applications in statistics...

    T. Pham-Giaet al. Testing the equality of several covariance matrices

    • ...This apparently paradoxical behavior is an example of how the algebra of random variables with uncertainty is quite different from the algebra of deterministic variables; see [56]...

    Jari P. Kaipioet al. The Bayesian Framework for Inverse Problems in Heat Transfer

    • ...Then, according to common practice in statistics [31], [32], we can accurately approximate ξ by a single chi-square variable to simplify the relevant analysis...

    Q. T. Zhang. Theoretical Performance and Thresholds of the Multitaper Method for Sp...

    • ...Products of random variables of various types of distributions have been investigated and their distributions determined in [2], [3]...
    • ...In their analysis, the authors of [13] have exploited the fact that a product of H-functions is also an H-function [3]...
    • ...In this paper, we employ the Mellin transform [3], [14], [15] to derive the expression for PDF of the product of Nakagami-m or Gamma distributed random variables...
    • ...The Mellin transform of a product of random variables is the product of the Mellin transforms of the individual random variables [3], [15]...
    • ...The Mellin transform has been employed previously in similar applications such as determining the PDF of product of beta, gamma and Rayleigh distributed variables [3], [7], [15]; however, the inverse Mellin transform and the resultant PDF is expressed in these contributions in the form of Meijer’s G function...
    • ...Next, according to the properties of the Mellin transform, the transform of the product of random variables is equal to the product of the Mellin transforms of the individual random variables [3], [15]...
    • ...In terms of probability theory, fX (x) in (1) denotes the PDF of the random variable X .T he inverse Mellin transform is defined as [3]...
    • ...Here, we use the property of Mellin transform whereby if M [f (t) ,z ]= F (z), then [3], [17]...
    • ...If we use the variable transform of (z +1 )/2 → z in the above equation, this does not affect the path of integration [3]...
    • ...From the Residue Theorem [3], [17], [19], we know that the complex integral at the right-hand side of the above equation can be computed by summing the residues of the integrand associated with all its poles...
    • ...of z = −j. Consequently, using the corresponding relationship characterizing the residues of multiple poles [3], [18], [19] and...

    Sohail Ahmedet al. Probability Distributions of Products of Rayleigh and Nakagami-m Varia...

    • ...Moreover, the generalized-�� distribution belongs to the (Fox) �� -function distributions family [18]...
    • ...We propose using the Gamma distribution due to the following reasons: (i) Gamma distribution is a Type-III Pearson distribution which is widely used in fitting distributions for positive RVs by matching the first and second moments [18], and (ii) since the PDF in (3) corresponds to the product of two Gamma RVs, one of the corresponding Gamma PDFs will dominate for large values of �� �� or �� �� [7]...

    Saad Al-Ahmadiet al. On the approximation of the generalized-Kappa distribution by a gamma ...

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