LAB1
... amazing result from probability theory that explains why the Gaussian distribution (aka "Bell Shaped Curve" or Normal distribution) applies to areas as far ranging as economics and physics. Below are two statements of the Central Limit Theorem (C.L.T.). I) "If an overall random variable is the sum o ...
... amazing result from probability theory that explains why the Gaussian distribution (aka "Bell Shaped Curve" or Normal distribution) applies to areas as far ranging as economics and physics. Below are two statements of the Central Limit Theorem (C.L.T.). I) "If an overall random variable is the sum o ...
Statistical Intervals
... Intervals Based on Nonnormal Population Distributions The one-sample t CI for µ is robust to small or even moderate departures from normality unless n is very small. By this we mean that if a critical value for 95% confidence, for example, is used in calculating the interval, the actual confidence ...
... Intervals Based on Nonnormal Population Distributions The one-sample t CI for µ is robust to small or even moderate departures from normality unless n is very small. By this we mean that if a critical value for 95% confidence, for example, is used in calculating the interval, the actual confidence ...
Probability Theory I
... In natural sciences, empirically one often observes that this quotient stabilizes around a certain number which we call probability of A. Then the additivity, for instance, is ...
... In natural sciences, empirically one often observes that this quotient stabilizes around a certain number which we call probability of A. Then the additivity, for instance, is ...
Learning Poisson Binomial Distributions
... The starting point of our algorithm for learning PBDs is a theorem of [DP11, Das08] that gives detailed information about the structure of a small ǫ-cover (under the total variation distance) of the space of all PBDs on n variables (see Theorem 4). Roughly speaking, this result says that every PBD i ...
... The starting point of our algorithm for learning PBDs is a theorem of [DP11, Das08] that gives detailed information about the structure of a small ǫ-cover (under the total variation distance) of the space of all PBDs on n variables (see Theorem 4). Roughly speaking, this result says that every PBD i ...
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.