On sequential occupancy problems

1981 ◽  
Vol 18 (02) ◽  
pp. 435-442 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Waiting Time ◽  
Random Variables ◽  
Characteristic Functions ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Y l, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

On sequential occupancy problems

1981 ◽  
Vol 18 (2) ◽  
pp. 435-442 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Waiting Time ◽  
Random Variables ◽  
Characteristic Functions ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

Limit theorems for some sequential occupancy problems

1983 ◽  
Vol 20 (03) ◽  
pp. 545-553 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Results ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.

Limit theorems for some sequential occupancy problems

1983 ◽  
Vol 20 (3) ◽  
pp. 545-553 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Results ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

Analytic solution of a finite dam governed by a general input

1974 ◽  
Vol 11 (01) ◽  
pp. 134-144 ◽  
Author(s):  
S. K. Srinivasan
Keyword(s):  
Total Duration ◽  
Random Variables ◽  
Rational Functions ◽  
Characteristic Functions ◽  
Dry Period ◽  
Laplace Transform Technique ◽  
Independent Families ◽  
Set Up ◽  
Input Form ◽  
Finite Dam

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.

A central limit theorem for exchangeable variates with geometric applications

1973 ◽  
Vol 10 (4) ◽  
pp. 837-846 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Sum Of Random Variables ◽  
Th Cell ◽  
Occupancy Problems

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

Some joint distributions in Bernoulli excursions

1994 ◽  
Vol 31 (A) ◽  
pp. 239-250
Author(s):  
Endre Csáki
Keyword(s):  
Random Walk ◽  
Random Variables ◽  
Characteristic Functions ◽  
Symmetric Random Walk ◽  
Joint Distributions ◽  
Recursion Formulas

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

scholarly journals The mean waiting time to a repetition

1983 ◽  
Vol 15 (01) ◽  
pp. 216-218
Author(s):  
Gunnar Blom
Keyword(s):  
Waiting Time ◽  
Mild Condition ◽  
Random Variables ◽  
Stationary Sequence ◽  
Mean Waiting Time ◽  
The Mean

Let X 1, X2, · ·· be a stationary sequence of random variables and E 1 , E 2 , · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence E j1 , E j2 , · ·· generated by the X's, the mean waiting time from an event, say E j1 , to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.

Poisson recurrence times

1967 ◽  
Vol 4 (03) ◽  
pp. 605-608
Author(s):  
Meyer Dwass
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variables ◽  
Recurrent Event ◽  
Recurrence Times ◽  
Image Position ◽  
Independent And Identically Distributed

Let Y1, Y 2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y 1 + … + Yn, n = 1,2,…, S 0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

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