Central Limit Theorem: Sampling Distribution.

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Presentation transcript:

Central Limit Theorem: Sampling Distribution. Jacek Wallusch _________________________________ Mathematical Statistics for International Business Lecture 6: Central Limit Theorem: Sampling Distribution.

a probability distribution of a statistic Getting Started ____________________________________________________________________________________________ definitions Statistic any real or vector-valued function of the observed random variables, describing a characteristic of the random variable Sampling Distribution a probability distribution of a statistic Sample a sequence of independent identically distributed random variables with a common distribution function or density Population distance measured in s.d. units a collection of all independent identically distributed random variables with a common distribution function or density Mathematical Statistics: 6

normal density function is symmetric with respect to the mean value Normal Distribution ____________________________________________________________________________________________ probability Probability, mean value and standard deviation normal density function is symmetric with respect to the mean value distance between mean value and any point at the normal curve: distance measured in s distance measured in s.d. units Mathematical Statistics: 6 Conversion formula z

Standard Normal Distribution ____________________________________________________________________________________________ probability Probability, mean value and standard deviation Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 6 Do not confuse z with the z-score

SND ____________________________________________________________________________________________ central limit theorem Assumptions: sequence of independent and identically distributed variables central moments, mean value and variance, defined as Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 6 mean value and variance are both FINITE

SND ____________________________________________________________________________________________ central limit theorem Assumptions: define additionally where Theorem states that: the distribution of UT converges to a standard normal distribution function as T approaches infinity Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 6 mean value of a large number of iid random variables is approximately normally distributed

Not everything lies within 2 standard deviation of the mean value Normal Distribution ____________________________________________________________________________________________ beware of misuse Not everything lies within 2 standard deviation of the mean value Probability distribution and uncertainty and risk – this topic will be reconsidered soon Do not follow blindly Mathematical Statistics: 6 ask your doctor about the normal distribution

CLT in Practice ____________________________________________________________________________________________ probability Calculate the probability MS Excel formula: standard normal distribution Probability distribution and uncertainty and risk – this topic will be reconsidered soon use the value of z, and subtract it from 1 Mathematical Statistics: 6 Do not confuse z with the z-score

CLT in Practice ____________________________________________________________________________________________ examples Exercises: 1. Use the data for EUR and USD, calculate the probability of obtaining different aims; 2. Use the data on expected salaries, calculate the probability that a female student claims to earn more than an average male student; Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 6 Do not confuse z with the z-score

CLT in Practice ____________________________________________________________________________________________ examples Helpful formulas: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 6 Do not confuse z with the z-score