# Some Characterizations of The Exponential Family

### Abstract

This paper introduces some characterizations concerning the exponential family. Recurrence relation between two consecutive conditional moments of h(z) given x<z<y is presented. In addition, an expression of V[h(Z]x<Z< y)as well as a closed form of E[h^{r}(Z)x<Z< y] in terms of the failure rate and the reversed failure rate is deduced. Finally, the left r^{th} truncated moment of h(Y_{k}) ( where Y_{k} is the K^{th} order statistic) is expressed in terms of a polynomial, h(-) , of degree r. Some results concerning the exponentiated Pareto, exponentiated Weibull, the Modified Weibull, Weibull, generalized exponential, Linear failure rate,1^{st} type Pearsonian distributions, Burr, power and the uniform distributions are obtained as special cases.

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### References

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