Consider the multicolored urn model where after every draw, balls of the different colors are added to the urn in proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three and four color urns, we derive the asymptotic behavior of linear combinations of number of balls. In particular, we show that certain li...

A classical problem in statistics is estimating the expected coverage of a sample, which has had applications in gene expression, microbial ecology, optimization, and even numismatics. Here we consider a related extension of this problem to random samples of two discrete distributions. Specifically, we estimate what we call the dissimilarity probability of a sample, i.e., the probability of a d...

2. Polya’s urn. Suppose we have an urn containing one red ball and one blue ball. We draw a ball at random from the urn. If it is red, we put the drawn ball plus another red ball into the urn. If it is blue, we put the drawn ball plus another blue ball into the urn. We then repeat this process. At the n-th stage, we draw a ball at random from the urn with n+1 balls, note its color, and put the ...

A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes balls are randomly drawn from urn with uncertain numbers of colored balls. This means the essentially intricate simultaneous randomness and epistemic uncertainty, this core discussed in paper. Since practical usually processed environment, solution for equivalent necessary. Suppose contain...

Abstract We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on bounded region typically known as swallow tail. will explicitly compute coefficients three-term recurrence relations generated by these QBD and conditions under we can produce families discrete-time processes. Final...

Consider the situation in which subjects arrive sequentially in a clinical trial. There are K possibly unrelated treatments. Suppose balls in an urn are labeled with treatments. When a subject arrives, a ball is drawn randomly from the urn, the subject receives the treatment indicated on the ball and the ball is returned to the urn. If the treatment is successful, one ball of the same type is a...