UDC 519.2
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 stojk@ukr.net
|
2 Institute of Physics and Technology of the National Technical University “Igor Sikorsky Kyiv Polytechnic Institute,” Kyiv, Ukraine
lusi.kovalchuk@gmail.com
|
EXACT ESTIMATIONS FOR SOME LINEAR FUNCTIONALS OF UNIMODAL DISTRIBUTION
FUNCTIONS UNDER INCOMPLETE INFORMATION
Abstract. Exact estimations are found for the probability that a non-negative unimodal random
variable μ gets in the interval (m – σμ,
m + σμ) where the mode m coincides
with fixed first moment
and σμ2
is fixed variance of random variable μ.
Also, a brief important auxiliary information is given with examples, statements, and author’s notations, which simplify obtaining the main result.
The results of this study may be useful in evaluating the probability of hitting the projectile zone when aimed shooting.
Keywords: linear functionals of unimodal distribution functions and their extremal values, transformation of Johnson–Rogers, exact generalized Chebyshoff inequalities for unimodal distribution functions
.
FULL TEXT
REFERENCES
- Karlin S., Stadden V. Chebyshev systems and their application in analysis and statistics [Russian translation]. Moscow: Nauka, 1976. 568 p.
- Mulholland H.P., Rogers C.A. Representation theorems for distribution functions. Proc. London Math. Soc. 1958. Vol. 8, N 3. P. 177–223.
- Johnson N.L., Rogers C.A. The moment problem for unimodal distribution. Ann. Math. Stat. 1951. Vol. 22. P. 433–439.
- Kleinrock L. Queuing Theory [Russian translation]. Moscow: Mashinostroyeniye, 1979. 432 p.
- Krein M.G., Nudelman A.A. Markov moment problem and extremal problems [in Russian]. Moscow: Nauka, 1973. 551 p.
- Stoykova L.S. A necessary and sufficient condition for the extremum of the Lebesgue–Stieltjes integral in the class of distributions. Kibernetika. 1990. Vol. 1. P. 72–75.
- Stoykova L.S. Generalized Chebyshev inequalities and their application in the mathematical theory of reliability. Kibernetika i sistemnyj analiz. 2010. Vol. 3. P. 139–143.
- Stoykova L.S. The greatest exact lower limit of the probability of system failure in a special time interval with incomplete information about the time distribution function before the system failure. Kibernetika i sistemnyj analiz. 2017. Vol. 53, N 2. P. 65–73.
- Ventzel E.S., Ovcharov L.A. Probability theory [in Russian]. Moscow: Nauka, 1973. 365 p.
