UDC 519.2
ACCURATE ESTIMATES OF THE PROBABILITY OF A RANDOM VARIABLE
FALLING INTO A GIVEN INTERVAL IN ONE CLASS OF UNIMODAL DISTRIBUTIONS
Abstract. A class A of unimodal distribution functions is considered, for which only the first two moments
and the mode m, which coincides with the first moment, are known. In this class, accurate estimates of the probability
of a random variable μ falling into the interval (0, 2m) are found.
Defined supplemental parameters α η and α μ ,
changing from 0 to ∞,
guide the entire process of solving the problem: extremal distribution functions are found that give accurate estimates
to the corresponding functional; highlighting the areas of existence of these distribution functions with numbers;
the problem turns out to be solved for all possible values of the mean square
deviations σ μ of the random variable μ.
Unimodal extremal distribution functions from class A were found in an explicit form.
Keywords: exact estimates, linear functionals, classes of unimodal distribution functions.
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