Chapter 7: Continuous Probability Distributions
... Recall from chapter 3 that any distribution can be standardized to have a mean of zero and a standard deviation of one. When µ = 0 and σ = 1, we call N (0, 1) the Standard Normal Distribution. µ=0 , σ=1 ...
... Recall from chapter 3 that any distribution can be standardized to have a mean of zero and a standard deviation of one. When µ = 0 and σ = 1, we call N (0, 1) the Standard Normal Distribution. µ=0 , σ=1 ...
Full text
... see [2, 3, 5]. There is a large set of literature addressing generalized Zeckendorf decompositions, these include[1, 8, 10, 11, 12, 13, 14, 15, 16, 17, 25, 26] among others. However, all of these results only hold for PLR Sequences. In this paper, we extend the results on Gaussian behavior and avera ...
... see [2, 3, 5]. There is a large set of literature addressing generalized Zeckendorf decompositions, these include[1, 8, 10, 11, 12, 13, 14, 15, 16, 17, 25, 26] among others. However, all of these results only hold for PLR Sequences. In this paper, we extend the results on Gaussian behavior and avera ...
The Circle Method
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
(pdf)
... picked number M is prime or composite, one could thus imagine doing experiments, trying all the numbers a between 0 and M − 1 and checking whether aM ≡ a ( mod M ); if there is one a that fails this congruence test, then M is composite. This looks like a nice algorithm to prove that M is composite, ...
... picked number M is prime or composite, one could thus imagine doing experiments, trying all the numbers a between 0 and M − 1 and checking whether aM ≡ a ( mod M ); if there is one a that fails this congruence test, then M is composite. This looks like a nice algorithm to prove that M is composite, ...
Modeling Continuous Variables
... Mathematical model – An abstraction and, therefore, a simplification of a real situation, one that retains the essential features. Real situations are usually much to complicated to deal with in all their details. ...
... Mathematical model – An abstraction and, therefore, a simplification of a real situation, one that retains the essential features. Real situations are usually much to complicated to deal with in all their details. ...
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.