Category: Statistics
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Convergence of Random Variables
1. Introduction There are two main ideas in this article. Law of Large Numbers: This states that the mean of the sample converges in probability to the distribution mean as increases. Central Limit Theorem: This states that the distribution of the sample mean converges in distribution to a normal distribution as increases. 2. Types of…
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Expectation of Random Variables
1. Expectation of a Random Variable The expectation of a random variable is the average value of . Definition 1 The expectation, mean or first moment of is defined to be The following notations are also used. Theorem 2 The Rule of the Lazy Statistician: Let , then the expectation of Y is 2. Properties…
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Random Variables
1. Introduction Definition 1 Random Variable: A random variable is a mapping which assigns real numbers to outcomes in . 2. Distribution Functions Definition 2 Distribution Function: Given a random variable , the cumulative distribution function (also called the \textsc{cdf}) is a function defined by: Theorem 3 Let have \textsc{cdf} and let have \textsc{cdf} .…
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Frequentists and Bayesians
1. Interpretations of Probability 1.1. Bayesians and Frequentists There are two possible ways to interpret the meaning of probability. 1.2. Maps and Territories Here ‘territory’ refers to the world as it exists or the reality as it is. The map refers to our model of the world or the way we see and interpret it.…
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Probability
1. Introduction Probability is the mathematical language for quantifying uncertainty. 2. Sample Space and Events The setup begins with an experiment being conducted. It can have a number of outcomes. The following are then defined: Definition 1 Sample Space: The sample space is the set of all possible outcomes. Definition 2 Realizations, Sample Outcomes or…