scholarly journals Discussion on D.C.M. Dickson & H.R. Waters Multi-Period Aggregate Loss Distributions for a Life Portfolio

1999 ◽  
Vol 29 (2) ◽  
pp. 311-314 ◽  
Author(s):  
Bjørn Sundt
Keyword(s):  
Positive Integer ◽  
Probability Function ◽  
The Other ◽  
Positive Probability ◽  
Probability Functions ◽  
Counting Distribution ◽  
Aggregate Loss ◽  
Image Position ◽  
Bold Face ◽  
Compound Probability

In the present discussion we point out the relation of some results in Dickson & Waters (1999) to similar results in Sundt (1998a, b).We shall need some notation. For a positive integer m let m be the set of all m × 1 vectors with positive integer-valued elements and m+ = m ~ {0}. A vector will be denoted by a bold-face letter and each of its elements by the corresponding italic with a subscript denoting the number of the elements; the subscript · denotes the sum of the elements. Let m0 be the class of probability functions on m with a positive probability at 0 and m+ the class of probability functions on m+. For j = 1,…, m we introduce the m × 1 vector ej where the jth element is one and all the other elements zero. We make the convention that summation over an empty range is equal to zero.Let g ∈ m0 be the compound probability function with counting distribution with probability function v ∈ 10 and severity distribution with probability function h ∈ m+; we shall denote this compound probability function by v V h. Sundt (1998a) showed thatwhere φv denotes the De Pril transform of v, given byMotivated by (2) Sundt (1998a) defined the De Pril transform φg of g byThis defines the De Pril transform for all probability functions in m0. Insertion of (2) in (3) givesand by solving φg(X) we obtainSundt (1998a) studies the De Pril transform defined in this way and found in particular that it is additive for convolutions.

scholarly journals Nonconstant Sum Red-And-Black Games with Bet-Dependent Win Probability Function

2007 ◽  
Vol 44 (2) ◽  
pp. 547-553 ◽  
Author(s):  
Laura Pontiggia
Keyword(s):  
Probability Function ◽  
Positive Probability ◽  
Probability Functions

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.

Nonconstant Sum Red-And-Black Games with Bet-Dependent Win Probability Function

2007 ◽  
Vol 44 (02) ◽  
pp. 547-553 ◽  
Author(s):  
Laura Pontiggia
Keyword(s):  
Probability Function ◽  
Positive Probability ◽  
Probability Functions

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.

Representation of tiled matrix rings as full matrix rings

1989 ◽  
Vol 105 (1) ◽  
pp. 67-72 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Positive Integer ◽  
Jacobson Radical ◽  
Integral Domain ◽  
Matrix Ring ◽  
The Other ◽  
Matrix Rings ◽  
Full Matrix ◽  
Image Position ◽  
Full Matrix Ring ◽  
Unique Maximal Ideal

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

scholarly journals Recursive Evaluation of Some Bivariate Compound Distributions

1999 ◽  
Vol 29 (2) ◽  
pp. 315-325 ◽  
Author(s):  
Raluca Vernic
Keyword(s):  
Probability Function ◽  
Bivariate Distribution ◽  
Compound Distributions ◽  
Counting Distribution ◽  
Image Position ◽  
Recursive Evaluation

AbstractIn this paper we consider compound distributions where the counting distribution is a bivariate distribution with the probability function (Pn1,n2)n1,n2≥0 that satisfies a recursion in the formWe present an algorithm for recursive evaluation of the corresponding compound distributions and some examples of distributions in this class.

scholarly journals Two New Models for the Two-Person Red-And-Black Game

2010 ◽  
Vol 47 (01) ◽  
pp. 97-108 ◽  
Author(s):  
May-Ru Chen ◽  
Shoou-Ren Hsiau
Keyword(s):  
Nash Equilibrium ◽  
Probability Function ◽  
Positive Probability ◽  
Probability Functions ◽  
New Models

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

scholarly journals Two New Models for the Two-Person Red-And-Black Game

2010 ◽  
Vol 47 (1) ◽  
pp. 97-108 ◽  
Author(s):  
May-Ru Chen ◽  
Shoou-Ren Hsiau
Keyword(s):  
Nash Equilibrium ◽  
Probability Function ◽  
Positive Probability ◽  
Probability Functions ◽  
New Models

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

scholarly journals Nonconstant Sum Red-And-Black Games with Bet-Dependent Win Probability Function

2007 ◽  
Vol 44 (02) ◽  
pp. 547-553
Author(s):  
Laura Pontiggia
Keyword(s):  
Probability Function ◽  
Positive Probability ◽  
Probability Functions

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.

On carnap and popper probability functions

1979 ◽  
Vol 44 (3) ◽  
pp. 369-373 ◽  
Author(s):  
Hugues Leblanc ◽  
Bas C. van Fraassen
Keyword(s):  
Probability Function ◽  
Propositional Calculus ◽  
Binary Function ◽  
Probability Functions ◽  
Image Position

With PC understood to be the propositional calculus of [3], call a binary function Pr from the wffs of PC to the reals a Carnap (probability) function if it meets requirements A1—A5 in Table I (with ‘⊢…’ short in A3—A4 for ‘… is a tautology’), and call the function a Popper (probability) function if it meets requirements Bl—B6 there:Leblanc established in [3] that every Carnap function is a Popper one, and he tendered proof of the converse. As reported by Stalnaker in [5], the proof unfortunately was incomplete, a mishap due to Leblanc's abbreviating ‘Pr(A, ∼ A) = 1’ as ‘⊢ A’ when Pr is a Popper function. Borrowing from [4], Leblanc did show, as Harper notes in [2, footnote 17], that(1) If Pr is a Popper function and A is a tautology, then Pr(A, ∼ A) = 1. He did not, however, show that(2) If Pr is a Popper function and Pr(A, ∼ A) = 1, then A is a tautology.Nor could he have done so: (2) is false, as the simplest of counterexamples shows. Denied (2), Leblanc had no hope of proving that every Popper function is a Carnap one: of a Popper function Pr it is easily ascertained that Pr meets requirement A4 if and only if Pr(A, ∼ A) = 1 just in case A is a tautology.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals The Dissociation of the Compound of Iodine and Thio-urea

1904 ◽  
Vol 24 ◽  
pp. 233-239 ◽  
Author(s):  
Hugh Marshall
Keyword(s):  
Nitric Acid ◽  
General Formula ◽  
Free Acid ◽  
The Other ◽  
Image Position

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

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