939,330 research outputs found
A random number generator for continuous random variables
A FORTRAN 4 routine is given which may be used to generate random observations of a continuous real valued random variable. Normal distribution of F(x), X, E(akimas), and E(linear) is presented in tabular form
Rounding of continuous random variables and oscillatory asymptotics
We study the characteristic function and moments of the integer-valued random
variable , where is a continuous random variables.
The results can be regarded as exact versions of Sheppard's correction. Rounded
variables of this type often occur as subsequence limits of sequences of
integer-valued random variables. This leads to oscillatory terms in asymptotics
for these variables, something that has often been observed, for example in the
analysis of several algorithms. We give some examples, including applications
to tries, digital search trees and Patricia tries.Comment: Published at http://dx.doi.org/10.1214/009117906000000232 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Maximum-entropy probability distributions under Lp-norm constraints
Continuous probability density functions and discrete probability mass functions are tabulated which maximize the differential entropy or absolute entropy, respectively, among all probability distributions with a given L sub p norm (i.e., a given pth absolute moment when p is a finite integer) and unconstrained or constrained value set. Expressions for the maximum entropy are evaluated as functions of the L sub p norm. The most interesting results are obtained and plotted for unconstrained (real valued) continuous random variables and for integer valued discrete random variables. The maximum entropy expressions are obtained in closed form for unconstrained continuous random variables, and in this case there is a simple straight line relationship between the maximum differential entropy and the logarithm of the L sub p norm. Corresponding expressions for arbitrary discrete and constrained continuous random variables are given parametrically; closed form expressions are available only for special cases. However, simpler alternative bounds on the maximum entropy of integer valued discrete random variables are obtained by applying the differential entropy results to continuous random variables which approximate the integer valued random variables in a natural manner. All the results are presented in an integrated framework that includes continuous and discrete random variables, constraints on the permissible value set, and all possible values of p. Understanding such as this is useful in evaluating the performance of data compression schemes
The Foster-Hart Measure of Riskiness for General Gambles
Foster and Hart proposed an operational measure of riskiness for discrete
random variables. We show that their defining equation has no solution for many
common continuous distributions including many uniform distributions, e.g. We
show how to extend consistently the definition of riskiness to continuous
random variables. For many continuous random variables, the risk measure is
equal to the worst--case risk measure, i.e. the maximal possible loss incurred
by that gamble. We also extend the Foster--Hart risk measure to dynamic
environments for general distributions and probability spaces, and we show that
the extended measure avoids bankruptcy in infinitely repeated gambles
Stein approximation for functionals of independent random sequences
We derive Stein approximation bounds for functionals of uniform random
variables, using chaos expansions and the Clark-Ocone representation formula
combined with derivation and finite difference operators. This approach covers
sums and functionals of both continuous and discrete independent random
variables. For random variables admitting a continuous density, it recovers
classical distance bounds based on absolute third moments, with better and
explicit constants. We also apply this method to multiple stochastic integrals
that can be used to represent U-statistics, and include linear and quadratic
functionals as particular cases
Absolutely continuous representations of random variables
We study representations of a random variable as an integral of an
adapted process with respect to the Lebesgue measure. The existence of such
representations in two different regularity classes is characterized in terms
of the quadratic variation of (local) martingales closed by
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