Math 319 Solutions to Homework 8
... Problem 1. (a) Give an example of a sequence that is bounded above but not bounded below and that has a convergent subsequence. There are many examples. eg xn = 1 if n is even and xn = −n if n is odd. (b) Explain how √ to construct a monotonic increasing sequence of rational numbers that converges t ...
... Problem 1. (a) Give an example of a sequence that is bounded above but not bounded below and that has a convergent subsequence. There are many examples. eg xn = 1 if n is even and xn = −n if n is odd. (b) Explain how √ to construct a monotonic increasing sequence of rational numbers that converges t ...
Review
... Z-scores for normally distributed variables are also normally distributed, but with mean 0 and standard deviation 1 z ~ N(0, 1) Z-scores for a variable with some other distribution (right skewed, uniform, etc.) will follow the same shape as the original distribution, but with mean 0 and standard dev ...
... Z-scores for normally distributed variables are also normally distributed, but with mean 0 and standard deviation 1 z ~ N(0, 1) Z-scores for a variable with some other distribution (right skewed, uniform, etc.) will follow the same shape as the original distribution, but with mean 0 and standard dev ...
The standard deviation as a descriptive statistic
... This is part of a general philosophy of an integrated approach to teaching descriptive statistics in secondary schools. Emphasising the interpretation of various summary statistics, like the histogram, the mean, the SD, the normal distribution, and others, in terms of data sets, will introduce stati ...
... This is part of a general philosophy of an integrated approach to teaching descriptive statistics in secondary schools. Emphasising the interpretation of various summary statistics, like the histogram, the mean, the SD, the normal distribution, and others, in terms of data sets, will introduce stati ...
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.