81,378 research outputs found

### On the Generalized Poisson Distribution

The Generalized Poisson Distribution (GPD) was introduced by Consul and Jain (1973). However, as remarked by Consul (1989), "It is very difficult to prove by direct summation that the sum of all the probabilities is unity". We give a shorter and more elegant proof based upon an application of Euler's classic difference lemma.Comment: 3 page

### On the Confidence Interval for the parameter of Poisson Distribution

The possibility of construction of continuous analogue of Poisson distribution with the search of bounds of confidence intervals for parameter of Poisson distribution is discussed. Also, in the article is shown that the true value of a parameter of Poisson distribution for the observed value $\hat x$ has Gamma distribution with the scale parameter, which is equal to one, and the shape parameter, which is equal to $\hat x$. The results of numerical construction of confidence intervals are presented.Comment: 12 pages (1 LaTeX file), 3 eps files (figures), references to Sections are correcte

### A Generalization of the Exponential-Poisson Distribution

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the $i$th order statistic. We derive the $r$th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the R\'enyi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented

### The median of a jittered Poisson distribution

Let $N_\lambda$ and $U$ be two independent random variables respectively distributed as a Poisson distribution with parameter $\lambda >0$ and a uniform distribution on $(0,1)$. This paper establishes that the median, say $M$, of $N_\lambda+U$ is close to $\lambda +1/3$ and more precisely that $M-\lambda-1/3=o(\lambda^{-1})$ as $\lambda\to \infty$. This result is used to construt a very simple robust estimator of $\lambda$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq 10^9$)
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