The random walk associated by the game of roulette

1981 ◽  
Vol 18 (4) ◽  
pp. 931-936 ◽  
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (04) ◽  
pp. 931-936
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier

1969 ◽  
Vol 6 (01) ◽  
pp. 224-226
Author(s):  
J.C. Hardin ◽  
A.L. Sweet
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Absorption Probabilities ◽  
Explicit Expressions

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier

1969 ◽  
Vol 6 (1) ◽  
pp. 224-226 ◽  
Author(s):  
J.C. Hardin ◽  
A.L. Sweet
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Absorption Probabilities ◽  
Explicit Expressions

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (01) ◽  
pp. 169-175
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b &gt; 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 &lt;m &lt;a) under different conditions.

scholarly journals A note on the random walk model arising in double diffusion

1982 ◽  
Vol 24 (2) ◽  
pp. 121-129 ◽  
Author(s):  
L. H. Liyanage ◽  
J. M. Hill ◽  
C. M. Gulati
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Diffusion Model ◽  
General Result ◽  
Generating Functions ◽  
Double Diffusion ◽  
Horizontal Position ◽  
Special Cases ◽  
One Step ◽  
The Right

AbstractThe discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

scholarly journals Standard Paths in Another Composition Poset

10.37236/1829 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jan Snellman
Keyword(s):  
Generating Functions ◽  
Hasse Diagram ◽  
Term Orders ◽  
The Right

Bergeron, Bousquet-Mélou and Dulucq [Ann. Sci. Math. Québec 19 (1995), 139–151] enumerated paths in the Hasse diagram of the following poset: the underlying set is that of all compositions, and a composition $\mu$ covers another composition $\lambda$ if $\mu$ can be obtained from $\lambda$ by adding $1$ to one of the parts of $\lambda$, or by inserting a part of size $1$ into $\lambda$. We employ the methods they developed in order to study the same problem for the following poset, which is of interest because of its relation to non-commutative term orders : the underlying set is the same, but $\mu$ covers $\lambda$ if $\mu$ can be obtained from $\lambda$ by adding $1$ to one of the parts of $\lambda$, or by inserting a part of size $1$ at the left or at the right of $\lambda$. We calculate generating functions for standard paths of fixed width and for standard paths of height $\le 2$.

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