Marsaglia, George
Overview
Works:  25 works in 28 publications in 1 language and 34 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author 
Publication Timeline
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Most widely held works by
George Marsaglia
The Unreasonable effectiveness of number theory by
Stefan A Burr(
Book
)
1 edition published in 1992 in English and held by 6 WorldCat member libraries worldwide
1 edition published in 1992 in English and held by 6 WorldCat member libraries worldwide
The radiation dose accumulated by blood diverted through a shunt by
Boeing Company(
Book
)
2 editions published in 1964 in English and held by 2 WorldCat member libraries worldwide
Modern techniques have made it possible to divert a portion of the circulating blood through a shunt outside the bodyfor example in heartlung machines, artificial kidneys, and coiled tubes where the blood may be exposed to radiation without danger to body tissues. There is some probability theory connected with such procedures, for the cells of the blood are thoroughly mixed in the body, and hence the number of times a blood cell passes through the shunt is a random variable. Several papers have been written to describe such systems by differential equations; this paper discusses the problem directly in terms of probability theory, finding the exact distribution of the number of times a blood cell has passed through the shunt and, in addition, a normal approximation which makes calculation of accumulated doses a matter of simple arithmetic. (Author)
2 editions published in 1964 in English and held by 2 WorldCat member libraries worldwide
Modern techniques have made it possible to divert a portion of the circulating blood through a shunt outside the bodyfor example in heartlung machines, artificial kidneys, and coiled tubes where the blood may be exposed to radiation without danger to body tissues. There is some probability theory connected with such procedures, for the cells of the blood are thoroughly mixed in the body, and hence the number of times a blood cell passes through the shunt is a random variable. Several papers have been written to describe such systems by differential equations; this paper discusses the problem directly in terms of probability theory, finding the exact distribution of the number of times a blood cell has passed through the shunt and, in addition, a normal approximation which makes calculation of accumulated doses a matter of simple arithmetic. (Author)
Small procedure for generating normal random variables by George Marsaglia(
Book
)
2 editions published in 1962 in English and held by 2 WorldCat member libraries worldwide
2 editions published in 1962 in English and held by 2 WorldCat member libraries worldwide
Oneline random number generators and their use in combinations by George Marsaglia(
Book
)
2 editions published in 1968 in English and held by 2 WorldCat member libraries worldwide
This is a discussion of some oneline random number generators, requiring a single FORTRAN instruction, together with a description of some short FORTRAN programs which mix several such generators. Evidence suggesting that the simple congruential generators are unsatisfactory continues to grow; one of the most promising alternatives is to mix several simple generators. These composite generators do better in various tests for randomness than do the simple congruential generators used at many computer centers
2 editions published in 1968 in English and held by 2 WorldCat member libraries worldwide
This is a discussion of some oneline random number generators, requiring a single FORTRAN instruction, together with a description of some short FORTRAN programs which mix several such generators. Evidence suggesting that the simple congruential generators are unsatisfactory continues to grow; one of the most promising alternatives is to mix several simple generators. These composite generators do better in various tests for randomness than do the simple congruential generators used at many computer centers
Moment crossings as related to density crossings(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
In this paper it is shown how the number of moment crossings of two symmetrical densities is related to the number of crossings of the densities. This generalizes a result of Fisher's recently proved by Finucan (1964) (A note on Kurtosis). (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
In this paper it is shown how the number of moment crossings of two symmetrical densities is related to the number of crossings of the densities. This generalizes a result of Fisher's recently proved by Finucan (1964) (A note on Kurtosis). (Author)
Optimal representation of a function as a linear combination of functions(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
This paper discusses the approximation of a given density function g(x) with a linear combination of densities f sub one (x), f sub two (x) ..., f sub n (x) in such a way that the approximation has maximum area but always lies below the given function
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
This paper discusses the approximation of a given density function g(x) with a linear combination of densities f sub one (x), f sub two (x) ..., f sub n (x) in such a way that the approximation has maximum area but always lies below the given function
Bounds for the rank of the sum of two matrices(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This paper gives upper and lower bounds on the rank of the sum of two matrices, and discusses their connection with the condition that rank be additive over a set of matrices. (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This paper gives upper and lower bounds on the rank of the sum of two matrices, and discusses their connection with the condition that rank be additive over a set of matrices. (Author)
Tables of the distribution of quadratic forms of ranks two and three by George Marsaglia(
Book
)
1 edition published in 1960 in English and held by 1 WorldCat member library worldwide
1 edition published in 1960 in English and held by 1 WorldCat member library worldwide
Elementary relations between uniform and normal distributions in the plane(
Book
)
1 edition published in 1962 in English and held by 1 WorldCat member library worldwide
1 edition published in 1962 in English and held by 1 WorldCat member library worldwide
Expressing the normal distribution with covariance matrix a + b in erms of one with covariance matrix a(
Book
)
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
Some problems involving circular and spherical targets(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This article is concerned with some problems which occur in certain tactical considerations: how should one place k circles (spheres) in the plane (3space) so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles (spheres). For k> 3 the problem seems hopeless, (except for certain special situations); the case for k = 3 is still unresolved and is being worked on by a number of investigators, and the case for k = 2 is solved completely in this paper. The results for k = 2 have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for k> 3. (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This article is concerned with some problems which occur in certain tactical considerations: how should one place k circles (spheres) in the plane (3space) so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles (spheres). For k> 3 the problem seems hopeless, (except for certain special situations); the case for k = 3 is still unresolved and is being worked on by a number of investigators, and the case for k = 2 is solved completely in this paper. The results for k = 2 have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for k> 3. (Author)
Random variables with independent binary digits(
Book
)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
Let X = .b1b2b3 ... be a random variable with independent binary digits bn taking values 0 or 1 with probabilities pn and qn. When does X have a density function. A continuous density function. A singular distribution. This note proves that the distribution X is singular is and only if the tail of the series Summation (log(pn/qn)) squared diverges, and that X has a density that is positive on some interval if and only if log(pn/qn) is a geometric sequence with ratio 1/2 for n greater than some k, and in that case the fractional part of (2 to the power k)X has an exponential density (increasing or decreasing with the uniform density a special case). It gives a sufficient condition for X to have a density, (Summation log (2 max (pn, qn))converges), but unless the tail of the sequence log(pn/qn) is geometric, ratio 1/2, the density is a weird one that vanishes at least once in every interval. (Author)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
Let X = .b1b2b3 ... be a random variable with independent binary digits bn taking values 0 or 1 with probabilities pn and qn. When does X have a density function. A continuous density function. A singular distribution. This note proves that the distribution X is singular is and only if the tail of the series Summation (log(pn/qn)) squared diverges, and that X has a density that is positive on some interval if and only if log(pn/qn) is a geometric sequence with ratio 1/2 for n greater than some k, and in that case the fractional part of (2 to the power k)X has an exponential density (increasing or decreasing with the uniform density a special case). It gives a sufficient condition for X to have a density, (Summation log (2 max (pn, qn))converges), but unless the tail of the sequence log(pn/qn) is geometric, ratio 1/2, the density is a weird one that vanishes at least once in every interval. (Author)
A method for producing random variables in a computer(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This paper describes a general procedure for producing random variables in a computer. The idea is to represent the required X in the form: X = C (M + U sub 1 + U sub 2 + U sub 3), some 97  99% of the time, where c is constant, M is a discrete random variable taking perhaps 8 values, and the U's are uniform random variables; the other 1  3% of the time. X is generated from a residual density by the rejection technique. These two methods for producing X are combined in the proper proportions in order that the resulting distribution for X be correct. The method is general in that it applies to a wide variety of density functions. Programs based on this procedure are very fast and require little computer storage space  typically, 18 constants and 20 instructions. (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
This paper describes a general procedure for producing random variables in a computer. The idea is to represent the required X in the form: X = C (M + U sub 1 + U sub 2 + U sub 3), some 97  99% of the time, where c is constant, M is a discrete random variable taking perhaps 8 values, and the U's are uniform random variables; the other 1  3% of the time. X is generated from a residual density by the rejection technique. These two methods for producing X are combined in the proper proportions in order that the resulting distribution for X be correct. The method is general in that it applies to a wide variety of density functions. Programs based on this procedure are very fast and require little computer storage space  typically, 18 constants and 20 instructions. (Author)
Onesided approximations by linear combinations of functions(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The paper discusses how to approximate a function g(x) from one side by a linear combination of functions f sub 1 (x) ..., f sub n (x) so as to minimize the area between the two. It discusses the problem as one of finding the point where a moving hyperplane last touches a convex set and an approximate procedure based on linear programming methods. It gives details of an algorithm for solving the problem, examples, and applications to Monte Carlo Theorygenerating random variables in a computer. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The paper discusses how to approximate a function g(x) from one side by a linear combination of functions f sub 1 (x) ..., f sub n (x) so as to minimize the area between the two. It discusses the problem as one of finding the point where a moving hyperplane last touches a convex set and an approximate procedure based on linear programming methods. It gives details of an algorithm for solving the problem, examples, and applications to Monte Carlo Theorygenerating random variables in a computer. (Author)
Still another method for producing normal variables in a computer(
Book
)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
A method for producing normal random variables in terms of uniform random variables U sub 1, U sub 2, U sub 3 ... If Y = U sub 1 + U sub 2 + U sub 3, then choosing one of the four random variables 2Y  3, (4Y  6)/3, (Y  7)/2 or (Y + 4)/2 in the proportions .8365, .11506, .00372 and .00372 will produce the required normal variate 98.6 per cent of the time. The other 1.4 per cent is devoted to the tail or a rejection technique in order that the composite be exact. The method leads to very fast computer programs which are easy to code and occupy little space in the computer. (Author)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
A method for producing normal random variables in terms of uniform random variables U sub 1, U sub 2, U sub 3 ... If Y = U sub 1 + U sub 2 + U sub 3, then choosing one of the four random variables 2Y  3, (4Y  6)/3, (Y  7)/2 or (Y + 4)/2 in the proportions .8365, .11506, .00372 and .00372 will produce the required normal variate 98.6 per cent of the time. The other 1.4 per cent is devoted to the tail or a rejection technique in order that the composite be exact. The method leads to very fast computer programs which are easy to code and occupy little space in the computer. (Author)
Conditional means and covariances of normal variables with singular covariance matrix(
Book
)
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
Regularities in congruential random number generators(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The paper suggests that points in nspace produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established previously for multiplicative congruential generators by showing that all the points fall in sets of relatively few parallel hyperplanes. The existence of many containing sets of parallel hyperplanes was easily established, but proof that the number of hyperplanes was small required a result of Minkowski from the geometry of numbersa symmetric, convex set of volume 2 to the nth power must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the course lattice structure of congruential generators. It gives a simple, selfcontained proof that points in nspace produced by the general congruential generator r sub (i+1) is identically eaual to a(r sub i) + b mod m must fall on a lattice with unitcell volume at least m to the power (n1). There is no restriction on a or b; this means that all congruential random number generators must be considered unsatisfactory in terms of lattices containing the points they produce, for a good generator of random integers should have an nlattice with unitcell volume 1. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The paper suggests that points in nspace produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established previously for multiplicative congruential generators by showing that all the points fall in sets of relatively few parallel hyperplanes. The existence of many containing sets of parallel hyperplanes was easily established, but proof that the number of hyperplanes was small required a result of Minkowski from the geometry of numbersa symmetric, convex set of volume 2 to the nth power must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the course lattice structure of congruential generators. It gives a simple, selfcontained proof that points in nspace produced by the general congruential generator r sub (i+1) is identically eaual to a(r sub i) + b mod m must fall on a lattice with unitcell volume at least m to the power (n1). There is no restriction on a or b; this means that all congruential random number generators must be considered unsatisfactory in terms of lattices containing the points they produce, for a good generator of random integers should have an nlattice with unitcell volume 1. (Author)
RATIOS OF NORMAL VARIABLES AND RATIOS OF SUMS OF UNIFORM VARIABLES(
)
1 edition published in 1964 in English and held by 0 WorldCat member libraries worldwide
The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the bivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio u sub 1 + ... + u sub n) (V sub 1 + ... + V sub m) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed
1 edition published in 1964 in English and held by 0 WorldCat member libraries worldwide
The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the bivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio u sub 1 + ... + u sub n) (V sub 1 + ... + V sub m) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed
A Note on the Compatibility of Distribution Functions(
)
1 edition published in 1953 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1953 in English and held by 0 WorldCat member libraries worldwide
THE CUMULATIVE EFFECT OF RANDOM LOSSES IN A TRANSMISSION LINE(
)
1 edition published in 1963 in English and held by 0 WorldCat member libraries worldwide
A study was made of the probability distribution of losses from reflections at discontinuities in a transmission line, on the assumption that the reflection coefficients at the discontinuities have random phase, and, in addition, random magnitudes associated with manufacturing uncertainties (tolerances)
1 edition published in 1963 in English and held by 0 WorldCat member libraries worldwide
A study was made of the probability distribution of losses from reflections at discontinuities in a transmission line, on the assumption that the reflection coefficients at the discontinuities have random phase, and, in addition, random magnitudes associated with manufacturing uncertainties (tolerances)
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Related Identities
 BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH
 Lagarias, J. C.
 Burr, Stefan A.
 Andrews, George E.
 BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
 BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA
 NORTH CAROLINA UNIV AT CHAPEL HILL Dept. of STATISTICS
 Bray, T. A.
 Florida State University Supercomputer Computations Research Institute
 MacLaren, M. Donald
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