369,929 research outputs found
Exact Probability Distribution versus Entropy
The problem addressed concerns the determination of the average number of
successive attempts of guessing a word of a certain length consisting of
letters with given probabilities of occurrence. Both first- and second-order
approximations to a natural language are considered. The guessing strategy used
is guessing words in decreasing order of probability. When word and alphabet
sizes are large, approximations are necessary in order to estimate the number
of guesses. Several kinds of approximations are discussed demonstrating
moderate requirements concerning both memory and CPU time. When considering
realistic sizes of alphabets and words (100) the number of guesses can be
estimated within minutes with reasonable accuracy (a few percent). For many
probability distributions the density of the logarithm of probability products
is close to a normal distribution. For those cases it is possible to derive an
analytical expression for the average number of guesses. The proportion of
guesses needed on average compared to the total number decreases almost
exponentially with the word length. The leading term in an asymptotic expansion
can be used to estimate the number of guesses for large word lengths.
Comparisons with analytical lower bounds and entropy expressions are also
provided
Probability distribution of the order parameter
The probability distribution of the order parameter is exploited in order to
obtain the criticality of magnetic systems. Monte Carlo simulations have been
employed by using single spin flip Metropolis algorithm aided by finite-size
scaling and histogram reweighting techniques. A method is proposed to obtain
this probability distribution even when the transition temperature of the model
is unknown. A test is performed on the two-dimensional spin-1/2 and spin-1
Ising model and the results show that the present procedure can be quite
efficient and accurate to describe the criticality of the system.Comment: 5 pages, 7 figures, to appear in Braz. J. Phys. 34, June 200
Probability distribution of drawdowns in risky investments
We study the risk criterion for investments based on the drawdown from the
maximal value of the capital in the past. Depending on investor's risk
attitude, thus his risk exposure, we find that the distribution of these
drawdowns follows a general power law. In particular, if the risk exposure is
Kelly-optimal, the exponent of this power law has the borderline value of 2,
i.e. the average drawdown is just about to divergeComment: 5 pages, 4 figures (included
The Probability Distribution for Non-Gaussianity Estimators
One of the principle efforts in cosmic microwave background (CMB) research is
measurement of the parameter fnl that quantifies the departure from Gaussianity
in a large class of non-minimal inflationary (and other) models. Estimators for
fnl are composed of a sum of products of the temperatures in three different
pixels in the CMB map. Since the number ~Npix^2 of terms in this sum exceeds
the number Npix of measurements, these ~Npix^2 terms cannot be statistically
independent. Therefore, the central-limit theorem does not necessarily apply,
and the probability distribution function (PDF) for the fnl estimator does not
necessarily approach a Gaussian distribution for N_pix >> 1. Although the
variance of the estimators is known, the significance of a measurement of fnl
depends on knowledge of the full shape of its PDF. Here we use Monte Carlo
realizations of CMB maps to determine the PDF for two minimum-variance
estimators: the standard estimator, constructed under the null hypothesis
(fnl=0), and an improved estimator with a smaller variance for |fnl| > 0. While
the PDF for the null-hypothesis estimator is very nearly Gaussian when the true
value of fnl is zero, the PDF becomes significantly non-Gaussian when |fnl| >
0. In this case we find that the PDF for the null-hypothesis estimator fnl_hat
is skewed, with a long non-Gaussian tail at fnl_hat > |fnl| and less
probability at fnl_hat < |fnl| than in the Gaussian case. We provide an
analytic fit to these PDFs. On the other hand, we find that the PDF for the
improved estimator is nearly Gaussian for observationally allowed values of
fnl. We discuss briefly the implications for trispectrum (and other
higher-order correlation) estimators.Comment: 10 pages, 6 figures, comments welcom
Exact probability distribution functions for Parrondo's games
We consider discrete time Brownian ratchet models: Parrondo's games. Using
the Fourier transform, we calculate the exact probability distribution
functions for both the capital dependent and history dependent Parrondo's
games. We find that in some cases there are oscillations near the maximum of
the probability distribution, and after many rounds there are two limiting
distributions, for the odd and even total number of rounds of gambling. We
assume that the solution of the aforementioned models can be applied to
portfolio optimization.Comment: 5 pages, 3 figure
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