Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0; 1) random variables (i.e., continuous random variables distributed uniformly over the interval (0; 1)). These random numbers are then transformed as needed...

A general parameter process defined by a continuum of random variables is not jointly measurable with respect to the usual product σ-algebra. For the case of independent random variables, a one-way Fubini extension of the product space was constructed in [11] to satisfy a limited form of joint measurability. For the general case we show that this extension exists if and only if there is a count...

We study statistical properties of the random variables Xσ(π), the number of occurrences of the pattern σ in the permutation π. We present two contrasting approaches to this problem: traditional probability theory and the “less traditional” computational approach. Through the perspective of the first one, we prove that for any pair of patterns σ and τ , the random variables Xσ and Xτ are jointl...

in a celebrated work by shao [13] several inequalities for negatively associated random variables were proved. in this paper we obtain some maximal inequalities for associated random variables. also we establish a maximal inequality for demimartingales which generalizes and improves the result of christofides [4].

We provide a new relaxation of conditions under which the inequality H(A|B,X) +H(A|B,Y ) 6 H(A|B) holds for jointly distributed random variables A,B,X, Y . Then we present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: Assume that edges of a bipartite graph are partitioned into K matchings so that for each pair (left vertex x, right verte...

In the absolutely continuous case, order statistics, record values and several other models of ordered random variables can be viewed as special cases of generalized order statistics, which enables a unified treatment of their theory. This paper deals with discontinuous generalized order statistics, continuing on the recent work of Tran (2006). Specifically, we show that in general neither re...

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulæ for the distribution of the sum of n non-identically distributed uniform random variables in both the continuo...