Category: Mathematics
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Parametric Inference
There are two methods of estimating . 1. Method of Moments It is a method of generating parametric estimators. These estimators are not optimal but they are easy to compute. They are also used to generate starting values for other numerical parametric estimation methods. Definition 1 Moments and Sample Moments: Suppose that the parameter has…
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Introduction to Statistical Inference
1. Introduction We assume that the data we are looking at comes from a probability distribution with some unknown parameters that control the exact shape of the distribution. Definition 1 Statistical Inference: It is the process of using given data to infer the properties of the distribution (for example the values of the parameters) which…
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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…
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Legendre Polynomials – Gram Schmidt for Function Spaces
If you are not familiar with function spaces, please read Function Spaces post before reading this. Sines and cosines form an orthonormal basis for the function space. If we want to use other functions as basis, we encounter the problem of their not being orthogonal to each other. For example, if we choose the simplest…
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Function Spaces and Fourier Series
We started with natural numbers to count discrete objects. Then we extended them to integers to have the closure property satisfied for the subtraction operator (or less formally, to be able to use numbers to describe profit and loss statements). Then we extended them to rationals to talk about ratios, then to reals to include…