2.
Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero. The series may diverge for other values of x. den{vertical-align:sub}. (2003).

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Let the corresponding quantile functions be QT(p), QR(p) and QY(p), where the quantile function is defined as QW(p) = inf {w : FW(w) ≥ p }, 0  p  1. For any real 0 ≤ R ≤ ∞, if the power series The radius of convergence of a power series is given by The radius of convergence is also given by The above is known as Hadamard Theorem. Then the power series distribution becomes, $$ \begin{eqnarray*} P(X=x) = \frac{a_x \theta^x}{f(\theta)}\\ = \frac{\theta^x}{x! e^\theta}\\ = \frac{e^{-\theta}\theta^x}{x!}, \; x=0,1,2,\cdots \end{eqnarray*} $$ which is the p.
Thus,
hence,
with equality if and only if v(x) = fT – R{Y} − PS(x) for all x except for a null measure set. of Binomial distribution with parameter $n$ and $p$. The new family was realized by compounding the T–R {Y} family of distribution and the power series family.

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These power series are also examples of Taylor series. A special member of the new family called the Gumbel–Weibull–{logistic}–Poisson (GUWELOP) distribution is defined and found to exhibit both unimodal and bimodal shapes. S. Observe that the pmf in (3) is truncated at zero and could be generalized to a zero-inflated one (Patil, [21]). [11]; Jones [12]; Cordeiro and de Castro [13]), method based on the composition of densities (Cooray and Ananda [14]) and the Transformed–Transformer method (Alzaatreh et al.

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f. D. ” American Mathematical Society, vol. Correspondence to
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Cockett, M. of Poisson distribution can be obtained by taking $f(\theta) = e^\theta$, we have $$ \begin{eqnarray*} M_X(t) = \frac{e^{\theta e^t}}{e^\theta}\\ =e^{\theta(e^t-1)}. The interval where it converges is called the interval of convergence.
Proof:
The proof follows by substituting U for p in (8), where U is a uniform random variable on the interval (0, 1). The m.
The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

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Some mathematical properties of the new class (including the moment generating function, moments and order statistics) are derived. D. Instant access to the full article PDF. Several mathematical properties of the new family were explored alongside the maximum likelihood method for the estimation of the parameters of the new family.

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, https://www. Estimation is addressed by the maximum likelihood method and the performance of the estimators assessed by a simulation study. Then the power series distribution becomes, $$ \begin{eqnarray*} P(X=x) = \frac{a_x\theta^x}{f(\theta)}\\ = \binom{n}{x}\frac{\theta^x}{(1+\theta)^n}\\ = \binom{n}{x}\bigg(\frac{\theta}{1+\theta}\bigg)^x\bigg(\frac{1}{1+\theta}\bigg)^{n-x} \end{eqnarray*} $$ Letting $p=\frac{\theta}{1+\theta}$, so $q=1-p = \frac{1}{1+\theta}$, we get $$ \begin{eqnarray*} P(X=x) =\binom{n}{x}p^xq^{n-x}\\ \quad x=0,1,2,\cdots, n,\\ \quad 0 p, q 1, \; p+q=1. From the theory of order statistics, the cdf of first order statistic X(1) for a given N = n is expressed as
Suppose N is a discrete random variable and follows the power series distribution in (3), the marginal cdf of X(1) can be written as
Thus, the cdf of the T–R {Y}–power series (T–R {Y}–PS) family of distributions is given by
A physical interpretation of the family of models in (4) is as follows: consider that the failure of a system, device, product, or component occurs due to the presence of an unknown number, say N, of initial defects of the same kind, which can be identifiable only after causing the failure and repaired perfectly. Received: 22 May 2019Accepted: 16 April 2020Published: 03 June 2020DOI: https://doi. [16]).

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If a power Related Site with radius of convergence r is given, one can consider analytic continuations of the series, i. .

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