Full text
... respect to the relatively prime pair (P, Q), where | P | > 1 and \Q\ = 1. If I(n) = UkUk+l...Uk+f^, where k e N\ {0} and / : N -» N is a strictly monotone increasing function, then the resulting series J^=\Ctn II{ri) converges to an irrational sum. Proof: We first establish convergence of the series ...
... respect to the relatively prime pair (P, Q), where | P | > 1 and \Q\ = 1. If I(n) = UkUk+l...Uk+f^, where k e N\ {0} and / : N -» N is a strictly monotone increasing function, then the resulting series J^=\Ctn II{ri) converges to an irrational sum. Proof: We first establish convergence of the series ...
Application to Stirling numbers
... Note that when we say that P is m-dimensional, we don’t just mean that it’s sitting in an m-dimensional space; it’s intrinsic dimension is m. In particular, in has non-empty interior as a subset of R^m. Let a(x_1,...,x_m) be some m-dimensional sequence whose terms satisfy some multidimensional LRE w ...
... Note that when we say that P is m-dimensional, we don’t just mean that it’s sitting in an m-dimensional space; it’s intrinsic dimension is m. In particular, in has non-empty interior as a subset of R^m. Let a(x_1,...,x_m) be some m-dimensional sequence whose terms satisfy some multidimensional LRE w ...
235_lecture9_080311a
... • Point estimate (sample statistic) gives us no idea of how close we might be to the true population parameter. • We want to be able to specify some interval around our point estimate that will have a high prob. of containing the true ...
... • Point estimate (sample statistic) gives us no idea of how close we might be to the true population parameter. • We want to be able to specify some interval around our point estimate that will have a high prob. of containing the true ...
finding areas with nonstandard normal
... • Don’t confuse z scores and areas. Remember z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 list z scores in the left columns and across the top row, but areas are found in the body of the table. • Choose the correct (right/left) side of th ...
... • Don’t confuse z scores and areas. Remember z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 list z scores in the left columns and across the top row, but areas are found in the body of the table. • Choose the correct (right/left) side of th ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.