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Cramér writes that the thesis has a reputation for being impossible to understand but, that looked at now, " one cannot help being struck by his ability to deal intuitively with concepts and methods that would have to wait another thirty years before being put on a rigorous foundation.
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Cramér and thesis
Away from the thesis Wold and Cramér did some joint work, their best known result being the Cramér – Wold theorem ( 1936 ).
Cramér and for
This can be used, for example, to compute the Cramér – Rao bound for parameter estimation in this setting.
Cramér remained close to Stockholm for most of his life.
During his tenure at Stockholm University, Cramér was a PhD advisor for 10 students, most notably Herman Wold and Kai Lai Chung.
Cramér remained an active contributor to his professional career for an additional 20 years.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they can be.
It was an opportune time, for Harald Cramér had been appointed Professor of Actuarial Mathematics and Mathematical Statistics.
After graduating in 1930 Wold worked for an insurance company ; he also did work on mortality data with Cramér and later designed a tariff for the insurance companies.
The Cramér – Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased.
The following is a proof of the general scalar case of the Cramér – Rao bound, which was described above ; namely, that if the expectation of is denoted by, then, for all,
This limitation, for example, can be found through the Cramér – Rao bound.
Comparing this to the variance of the sample mean ( determined previously ) shows that the sample mean is equal to the Cramér – Rao lower bound for all values of and.
Harald Cramér conjectured that the gap is always much smaller, ; if Cramér's conjecture is true, Legendre's conjecture would follow for all sufficiently large numbers.
Cramér and at
In the late 1920s, Cramér became interested in the field of probability, which at the time was not an accepted branch of mathematics.
During the years from 1961 to 1983, Cramér traveled throughout the United States and Europe to continue his research, making significant stops at Berkeley, Princeton, and at the Research Triangle Institute of North Carolina.
The Gram – Charlier A series diverges in many cases of interest — it converges only if F ( x ) falls off faster than exp (− x < sup > 2 </ sup >/ 4 ) at infinity ( Cramér 1957 ).
According to Harald Cramér, " Filip Lundberg's works on risk theory were all written at a time when no general theory of stochastic processes existed, and when collective reinsurance methods, in the present sense of the word, were entirely unknown to insurance companies.
Cramér and one
Cramér proved that in this model, the above conjecture holds true with probability one.
Cramér and by
Cramér knew that a radical change was needed in this field, and in a paper in 1926 said, " The probability concept should be introduced by a purely mathematical definition, from which its fundamental properties and the classical theorems are deduced by purely mathematical operations.
" Cramér mentions later work by Andrey Kolmogorov and William Feller but it was Cramér himself who developed Lundberg's ideas on risk and linked them to the emerging theory of stochastic processes.
Reprinted in The Collected Works of Harald Cramér edited by Anders Martin-Löf, 2 volumes Springer 1994.
Cramér and with
He started work on a PhD on stochastic processes with Cramér as supervisor.
* Harald Cramér ( 1976 ) Half a Century with Probability Theory: Some Personal Recollections, Annals of Probability, Vol.
Photograph of Rao with Harald Cramér in 1978
Cramér and would
Shortly after World War II, Cramér would go on to publish the influential Mathematical Methods of Statistics in 1946.
Dr. Cramér would eventually retire from the Swedish university system in 1961.
Cramér and on
Harald Cramér was born in Stockholm, Sweden on September 25, 1893.
Early on, Cramér was highly involved in analytic number theory.
Starting in 1950, Cramér took on the additional responsibility of becoming the President of Stockholm University.
In estimation theory and statistics, the Cramér – Rao bound ( CRB ) or Cramér – Rao lower bound ( CRLB ), named in honor of Harald Cramér and Calyampudi Radhakrishna Rao who were among the first to derive it, expresses a lower bound on the variance of estimators of a deterministic parameter.
For the Cramér – Rao inequality and the Rao – Blackwell theorem see the relevant entries on
* Harald Cramér ( 1969 ) Historical Review of Filip Lundberg's Work on Risk Theory, Skandinavisk Aktuarietidskrift ( Suppl.
Cramér also proved that the Riemann hypothesis implies a weaker bound of on the size of the largest prime gaps.
Cramér and rigorous
" Cramér took an interest in the rigorous mathematical formulation of probability in the work of French and Russian mathematicians such as Kolmogorov, Lévy, Bernstein, and Khinchin in the early 1930s.
Cramér and .
Another test statistic having this property is the Watson statistic, which is related to the Cramér – von Mises test.
Harald Cramér (; September 25, 1893 – October 5, 1985 ) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory.
Cramér also made significant development to the revolution in probability theory.
Cramér later wrote his careful study of the field in his Cambridge publication Random variables and probability distributions which appeared in 1937.
In 1929, Cramér was appointed to a newly created chair in Stockholm University, becoming the first Swedish professor of Actuarial Mathematics and Mathematical Statistics.
Cramér retained this position up until 1958.
Harald Cramér married Marta Hansson in 1918, and they remained together up until her death in 1973.
This is known as the Cramér model of the primes.
In mathematical statistics, Wold contributed the Cramér – Wold theorem characterizing the normal distribution and developed the Wold decomposition in time series analysis.
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