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Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

A generalized negative binomial (GNB) distribution with an additional parameter / has been obtained by using Lagrange's expansion. The parameter is such that both mean and variance tend to increase or decrease with an increase or decrease in its value but the variance increases or decreases faster than the mean. For/ 1/2, the mean and variance are approximately equal and so the GNB distribution resembles the Poisson distribution. When / 0 or 1, the GNB distribution reduces to the binomial or negative binomial distribution respectively. It has been shown that the generalized negative binomial distribution converges to a Poisson-type distribution in which the variance may be more or less than the mean, depending upon the value of a parameter. Expected frequencies have been calculated for a number of examples to show that the distribution provides a very satisfactory fit in different practical situations. Its convolution property together with other properties are quite interesting.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association and American Society for Quality are collaborating with JSTOR to digitize, preserve and extend access to Technometrics.A new generalization of the Poisson distribution, with two parameters X, and X2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as X2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of X\ and X2 are given. KEY WORDS

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