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Discrete Math. Appl. 2015; 25 (5):311–321 Aleksey D. Yashunsky On read-once transformations of random variables over nite elds Abstract: Transformations of independent randomvariables over a nite eld by read-once formulas are con- sidered. Subsets of probability distributions that are preserved by read-once transformations are constructed. Also we construct a family of distributions that may be arbitrarily closely approximated by a read-once com- bination of independent identically distributed random variables, whose distributions have no zero compo- nents

1 Introduction Some recurrence relations and identities on the distribution function ( df ) and probability density function ( pdf ) of order statistics, from independent and identically distributed ( iid ) random variables, are provided in the literature by many authors including Arnold et al . [ 1 ], David [ 2 ], Balasubramanian and Beg [ 3 ], and Reiss [ 4 ]. Furthermore, Arnold et al . [ 1 ], Gan and Bain [ 5 ], David [ 2 ], and Khatri [ 6 ] established the df and probability function ( pf ) of order statistics of iid discrete random variables. Corley

1 Introduction and preliminaries Since the concept of independence is fundamental in Probability Theory, Mathematical Statistics and their applications, various notions have been developed related to the independence of random variables. See, for example [ 7 : Sections 3 and 7]. The earliest of such notions are uncorrelatedness and correlation coefficient, both widely used in statistical analysis. For a brief history and their relation to the independence property, see [ 2 , 3 ]. An extension of the uncorrelatedness property for the powers of random variables

. [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990. [18] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990. [19] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990. [20] Hiroyuki Okazaki and Yasunari Shidama. Random variables and product of probability spaces. Formalized Mathematics, 21(1):33-39, 2013. doi:10.2478/forma-2013-0003. [21] Beata Padlewska. Families of sets. Formalized

References [1] ALAM, K.-SAXENA, K. M. L.: Positive dependent in multivariate distributions, Comm. Statist. Theory Methods 10 (1981), 1183-1196. [2] ASMUSSEN, S. et al: Stationarity detection in the initial transient problem, ACMTrans. Modeling Comput. Simulation 2 (1992), 130-157. [3] AZUMA, K.: Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), 357-367. [4] BENTKUS, V.: An inequality for large deviation probabilities of sums of bounded i.i.d. r.v, Lith. Math. J. 41 (2001), 144-153. [5] BENTKUS, V.: An inequality for tail

-167, 1990. [12] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics , 20( 1 ):1-5, 2012. doi:10.2478/v10037-012-0001-5. [13] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics , 1( 2 ):401-407, 1990. [14] Hiroyuki Okazaki. Probability on finite and discrete set and uniform distribution. Formalized Mathematics , 17( 2 ):173-178, 2009. doi:10.2478/v10037-009-0020-z. [15] Hiroyuki Okazaki and Yasunari Shidama. Probability on finite set and real-valued random variables. Formalized Mathematics , 17( 2 ):129

c© Heldermann Verlag Economic Quality Control ISSN 0940-5151 Vol 20 (2005), No. 2, 241 – 246 Ratio of Logistic and Bessel Random Variables Saralees Nadarajah and Samuel Kotz Abstract: The distribution of the ratio ∣∣X Y ∣∣ is derived when X and Y are logistic and Bessel function random variables distributed independently of each other. The distribution is of interest in econometrics, and ranking and selection problems. 1 Introduction For given random variables X and Y , the distribution of the ratio ∣∣X Y ∣∣ is of interest in econometrics, and ranking and

References [1] A. Akkurt, Z. Kaçar, H. Yildirim. Generalized fractional integral inequalities for continuous random variables. Journal of Probability and Statistics . Vol 2015. Article ID 958980, (2015), 1-7. [2] N. S. Barnett, P. Cerone, S.S. Dragomir and J. Roumeliotis. Some inequalities for the expectation and variance of a random variable whose PDF is n-time differentiable. J. Inequal. Pure Appl. Math. 1, (2000), 1-29. [3] N. S. Barnett, P. Cerone, S.S. Dragomir and J. Roumeliotis. Some inequalities for the dispersion of a random variable whose PDF is

Statistics & Decisions 23, 131–146 (2005) c© R. Oldenbourg Verlag, München 2005 Recursive random variables with subgaussian distributions Ralph Neininger Received: Mai 5, 2005; Accepted: September 23, 2005 Summary: We consider sequences of random variables with distributions that satisfy recurrences as they appear for quantities on random trees, random combinatorial structures and recursive algo- rithms. We study the tails of such random variables in cases where after normalization convergence to the normal distribution holds. General theorems implying

Random Oper. Stoch. Equ. 21 (2013), 21–35 DOI 10.1515/rose-2013-0002 © de Gruyter 2013 Distribution of random variable represented by binary fraction with two redundant digits 2 and 3 having the same distribution Mykola V. Pratsiovytyi and Oleg P. Makarchuk Communicated by Anatoly F. Turbin Abstract. We study the properties of the distribution of the random variable represented by binary fraction with two redundant digits 2 and 3. The problem on the Lebesgue type of the distribution of with identically distributed digits is completely solved. Keywords