5.5.3 Convergence in Distribution
... giving us an approximation for the variance of our estimator. Example (Approximate mean and variance) Suppose X is a random variable with Eµ X = µ 6= 0. If we want to estimate a function g(µ), a first-order approximation would give us g(X) = g(µ) + g 0 (µ)(X − µ). If we use g(X) as an estimator of g ...
... giving us an approximation for the variance of our estimator. Example (Approximate mean and variance) Suppose X is a random variable with Eµ X = µ 6= 0. If we want to estimate a function g(µ), a first-order approximation would give us g(X) = g(µ) + g 0 (µ)(X − µ). If we use g(X) as an estimator of g ...
Week 1: Descriptive Statistics
... The nicotine content in a single cigarette of a particular brand is a random variable with mean = 0.8 mg and = 0.10 mg. If an individual smokes 5 packs of these cigarettes per week, what is the probability that the total amount of nicotine consumed in a week is at least 82 mg? ...
... The nicotine content in a single cigarette of a particular brand is a random variable with mean = 0.8 mg and = 0.10 mg. If an individual smokes 5 packs of these cigarettes per week, what is the probability that the total amount of nicotine consumed in a week is at least 82 mg? ...
Exam 3
... Be able to name all discrete and continuous probability distributions studied thus far. Know the mean and standard deviation of a standard normal random variable Be able to describe the sampling distribution of the mean Know under what conditions the Central Limit Theorem applies. Know when ...
... Be able to name all discrete and continuous probability distributions studied thus far. Know the mean and standard deviation of a standard normal random variable Be able to describe the sampling distribution of the mean Know under what conditions the Central Limit Theorem applies. Know when ...
Central limit theorem: X is the sample mean of a
... A random sample of size n from f(x) is a collection of n random variables X1, . . ., Xn such that X1, . . ., Xn are independent and each Xi has the probability distribution defined by f(x). In R, we can generate random samples from many frequently used f(x). Our Monte Carlo simulation will proceed a ...
... A random sample of size n from f(x) is a collection of n random variables X1, . . ., Xn such that X1, . . ., Xn are independent and each Xi has the probability distribution defined by f(x). In R, we can generate random samples from many frequently used f(x). Our Monte Carlo simulation will proceed a ...
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.