Random Variables Xi Research Articles

The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (Ω,B,P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX pq with the interpretation thatX pq (ω) is the distance betweenp andq at the “instant” ω. The probability of the eventJ pqr = {ω ∈ Ω:X pr (ω)≤X pq (ω)+X qr (ω)} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, “Statistical Metrics,” Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, “Statistical Metric Spaces,” Pacific J. Math.10 (1960), 313–334). It turns out that for e > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X <Y +Z) < e. This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J pqr ) = 1.

Inference ony&gt;y o by using a correlated variable: A three decision approach

In an extension of the two decision approach [Bauer, Scheiber andWohlzogen, 1975] a Bayes solution is aimed at for the three decisiony>yo,y≤yo or “no classification” on the basic of the measurement of a positively correlated random variableX, which can be measured more easily and/or with smaller expense. Assuming a bivariate normal distribution forX andY optimal decision regions for the measuredx are derived in the case of constant or exponentially increasing losses.

A supplement to the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ &gt; 0, the number of integers n such that |n−1 Σi=1nXi − μ| &gt; ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) &lt; ∞ if and only if EX12 &lt; ∞. Here it is shown that if EX12 &lt; ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.

A supplement to the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ &amp;gt; 0, the number of integers n such that |n −1 Σ i=1 n X i − μ| &amp;gt; ∊, N (∊), is finite a.s. It is known, furthermore, that EN (∊) &amp;lt; ∞ if and only if EX 1 2 &amp;lt; ∞. Here it is shown that if EX 1 2 &amp;lt; ∞ then ∊ 2 EN (∊) → var X 1 as ∊ → 0.

Some asymptotic results on multiple matching

We considers≦nrandomly chosen permutations of the numbers 1, 2, …,n, and write them under each other, thus forming ans×nmatrix, called “random-batch”. A rule, prescribing how many elements of a column may occur exactly one time, how many may occur exactly two times, etc., is called a fixpoint-structure. Assuming each possible permutation to be chosen with equal probability, the number of columns having a certain fixpoint-structureFis a random variableX(F). The limiting distribution ofX(F) forn→ ∞is considered for different cases ofs=s(n). The main result (Theorem 4) says, that the common distribution of a finite numbertof given fixpoint-structures tends to the product oftPoisson-laws,n→ ∞.

On the Decomposition of Moments by Conditional Moments

M\'any practical problems in applied statistics involve the determination of the mean and variance of a certain statistic. The usual procedure is to attempt to find the distribution, i.e. probability density function or characteristic function, of the statistic, and then, if successful, to carry out the appropriate integration or differentiation. In many practical situations, information about the distribution may not be complete, or the sample may not consist of independent and identically distributed random variables, but of random variables whose distributions differ and which are correlated. The method of decomposition by conditional moments which requires only the knowledge of certain conditional moments may in some cases be helpful. Let X1, . . . , Xk be random variables of dimensions n, ... ., n, respectively. Assume that the means and covariances of the conditional random variables Xi I (Xi+1, .. ., Xk) for i =1,...,k-1 as well as the unconditional random variable Xk are known. If Z, and Z2 are any txvo random variables which are functions of X1, . . , Xk then for j = 1, 2,

Branching and clustering models associated with the ‘lost-games distribution’

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi with probability generating functions gi(s), i = 1, 2, 3, if g3(s) = g1[g2(s)], we say that X3 is X1 generalized by X2 and write X3~ X1VX2.

Simultaneous Pairwise Linear Structural Relationships

Prompted by an example in the medical field we are led to consider linear structural relationships between all pairs of a set of p(>2) random variables Xi(i = 0, ... , p 1). This model has relevance in assessing the relative calibrations and relative accuracies of a set of p instruments, each designed to measure the same characteristic, on a common group of individuals. In such a situation we need to estimate the slope and intercept parameters in the linear structural relationships, and the error variances, as measures of calibration and accuracy, respectively, for the p instruments. In contrast to the case p = 2, no problem occurs of unideiltifiability of the parameters and the maximum likelihood approach is feasible. This is illustrated for the case p = 3, but rather simpler estimators are presented for p > 3, together with their asymptotic standard errors. The case p = 3 may be viewed as a new and useful form of instrumental variable approach to the difficult p = 2 situation. The theoretical results are used in an example involvilng different types of instrumenlts for measuring human lung capacity.

The Pareto optimality of ann-Company reinsurance treaty

Abstract Let n insurance companies each have differentiable utilities of money ui (·) (i = 1, …, n), where ui (m) is the value to Company i of m units of money, and let u′ i (·) be non-negative and non-increasing. Company i possesses an amount of capital Ci (including its policy-holders' premiums) as well as a portfolio of policies which subjects it to payment of the random variable Xi . A mutual reinsurance treaty among the n companies is an n-tuple of functions z = (z 1, …, zn ), where zi = zi (X 1, …, Xn ) represents the amount Company i shall pay upon occurrence of the claims X 1, …, Xn . Write X = (X 1, …, Xn ). Of course for all values of the Xi 's.

On a new property of partially balanced association schemes useful in psychometric structural analysis.

In an earlier paper [Psychometrika,31, 1966, p. 147], Srivastava obtained a test for the HypothesisH0 : Σ =α0Σ0 + ... +αlΣl, where Σi are known matrices,αi are unknown constants and Σ is the unknown (p ×p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p ×p) covariance matrix of a random variablex (withp components) the condition that Σ0, Σ1, ..., Σl form a commutative linear associative algebra and a certain vectorΘ, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where Σ0, Σ1, ..., Σl are the association matrices of a partially balanced association scheme.

On a property of the binomial distribution

In this paper we establish a property of the binomial distribution This property refers to the conditional distribution ofY forX=x in the bivariate discrete distribution of (X, Y). It has been shown that when this conditional distribution is the convolution of a random variableX 1 with the random variableX 2, then the form of the joint distribution is uniquely determined by the marginal distribution ofX and the distribution of the convoluteX 2. The distribution ofX 1 is uniquely determined and is of the binomial form. We have established an equation connecting the joint distribution of (X, Y) with the marginal ofX and distributions ofX 1 andX 2.

On the conditional expected value of contributions from a renewal process

The problem discussed in this paper arose from the study of the effects of unresolved resonances on neutron cross-sections, but it is considered here in more general terms.Events in a modified renewal process occur at successive intervals x1, x2, …, and the ith event has associated with it a parameter Γi. The random variables xi and Γi are all independent, and their probability distribution functions are known. Each event contributes to two quantities F(u) and G(v) measured at u and v respectively. The value of the total contribution of all events to G(v) is assumed to be known from observation: this paper gives formulae for the mean value of F(u) conditional on this known value of G(v).

On a method for generalizations of tchebycheff’s inequality

This paper contains a study of inequalities of the type of Tchebycheff’s, i.e., inequalities for the probability value of a given event in terms of known moments. Suppose that there is given a set of moments {μ 0=1,μ 1, ...,μ 2n } of a random variableX and letE be any given closed (or open) set in one dimension. The present work considers the criteria for the sharp upper and sharp lower bounds ofP r{X ∉E}*), and investigates the existence and properties of extremal distributions. General results are stated in section 2, with proofs in section 3. Some examples are given in the final section, where the setE is considered to be a discrete set in Example 1, an interval symmetrical about the mean in Example 2 with 2n=4, and two-dimensional cases are considered in Example 3.

A law of large numbers for abstract random variables

It was emphasized by Frechet [8] that in many of the practical applications, we get, as a result of a random experiment, an abstract random variableX belonging to a setX. Givenn determinations of the variableX, an arithmetic meanZ n is introduced in 2-. It reduces to the classical arithmetic mean, whenX is the set of real numbers. Moreover, if the random variableX is Gaussian, having a meanM(X), and belongs to a certain (4-) familyF, the introduced arithmetic meanZ n constitutes an exhausting estimation [5] of the meanM(X).