# On the Distribution of the Minimum and of the Maximum of a Random Number of I.I.D. Random Variables

• Moshe Shaked
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 17)

## Summary

Let Xi, i = 1,2,… be i.i.d. random variables with a common distribution function F(x). Let N be a positive integer-valued random variable, independent of the X’s, with a probability generating function ψ(u). Then G(x) = ψ(F(x)) and H(x) = 1−ψ(1−F(x)) are, respectively, the distributions of max(X1,…,XN) and of min(X1,…,XN). In sections 2 and 3 we derive some inequalities concerning F, G and H, their densities and their hazard rates. We also discuss there some cases in which the monotonicity of the hazard rate of F is preserved by G and H. In section 4 we give some solutions of a functional equation that appears as a condition in some of the theorems of section 3. We also characterize the geometric distributions by requiring the solutions to satisfy an additional assumption. By taking Xi and/or N as random vectors, we introduce in section 5 some methods of construction of multivariate distributions with desired marginals.

## Key Words

Series and parallel systems hazard rate random walks geometric distribution Jensen’s inequality functional equation multivariate distributions with desired marginals log-convex functions

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## References

1. [1]
Barlow, R. E. and Proschan, E. (1965). Mathematical Theory of Reliability. Wiley and Sons, New York.
2. [2]
Buhrman, J. M. (1973). Statistica Neerlandica 27, 125–126.
3. [3]
Dubey, D. S. (1969). Naval Res. Logist. Quart. 16, 37–40.Google Scholar
4. [4]
Dykstra, R. L., Hewett, J. E. and Thompson, W. A. (1973). Ann. Statist. 1, 674–681.
5. [5]
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 ( 3rd ed. ). John Wiley and Sons, Inc., New York.
6. [6]
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2 ( 2nd ed. ). John Wiley and Sons, Inc., New York.
7. [7]
Hardy, G. H., Littlewood, J. E. and Polya, G. (1952). Inequalities, (2nd ed.). Cambridge University Press.Google Scholar
8. [8]
Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics, Continuous Multivariate Distributions. John Wiley and Sons, Inc., New York.
9. [9]
Karlin, S. and McGregor, J. (1968a). Amer. Math. Soc. Trans. 132, 115–136.
10. [10]
Karlin, S. and McGregor, J. (1968b). Amer. Math. Soc. Trans. 132, 137–145.
11. [11]
Kuczma, M. (1968). Functional Equations in a Single Variable. Polish Scientific Publishers.Google Scholar
12. [12]
Malik, H. J. and Abraham, B. (1973). Ann. Statist. 1, 588–590.
13. [13]
Marshall, A. W. and Proschan, F. (1972). In Inequalities, O. Shisha (ed.), 3, 225–234.Google Scholar
14. [14]
Raghunandanan, K. andPatil, S. A. (1972). Statistica Neerlandica 26, 121–126.
15. [15]
Shaked, M. (1974). A concept of positive dependence for exchangeable random variables. Submitted for publication. (Abstracted in Bull. Inst. Math. Statist. 2, 218–219.)Google Scholar
16. [16]
Takahasi, K. (1965). Ann. Inst. Statist. Math. 17, 257–260.
17. [17]
Widder, D. V. (1941). The Laplace Transform. Princeton University Press.Google Scholar
18. [18]
van Zwet, W. R. (1964). Convex Transformation of Random Variables. Mathematical Center, Amsterdam.Google Scholar