Math
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Probability space, Sample space, Event, and Elementary eventMath/Probability 2020. 2. 5. 12:33
1. Overview Probability space or a probability triple $(\Omega ,F,P)$ is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly. Sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. Event is a set of outcomes of an e..
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Conditional expectationMath/Probability 2020. 2. 5. 11:18
1. Overview In probability theory, the conditional expectation of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur 2. Description 2.1 Formula ith two random variables, if the expectation of a random variable X is expressed conditional on another random variab..
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SVD, matrix inverse, and pseudoinverseMath/Linear algebra 2020. 1. 25. 10:36
1. Overview 2. Description 2.1 Inverse Full rank square matrix A Now I'm going to invert A which is fine we assume for the moment that A is an invertible matrix. So it's square and full rank. And of course, whatever operation you perform on one side of the equation must be repeated on the other side of the equation. So we apply the inverse to the right-hand side as well. Now we know that each of..
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Least-squares for model fittingMath/Linear algebra 2020. 1. 24. 14:20
1. Overview There are uncountable dynamics and processes and individuals with uncountable and mind-bogglingly complex interactions. So how can we possibly make sense of all of this complexity? The answer is we can't :( So instead we generate simple models and we fit models to data using linear least squares modeling and that is the goal of this section of the course. the idea of building models ..
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Linear Algebra FeaturesMath/Linear algebra 2020. 1. 23. 18:03
1. Overview Summarize Terminologies and features 2. Description 2.1 Matrix multiplications It doesn't matter if you're multiplying $A^{T}A$ or $AA^{T}$ both results will produce not only a Square matrix but a Symmetric matrix. 2.1.1 Characteristic equation The characteristic equation is the equation that is solved to find a matrix's eigenvalues, also called the characteristic polynomial. For a g..
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Statistics and ProbabilityMath/Probability 2020. 1. 15. 22:12
1. Overview Statistics focuses predominantly on samples and incomplete data. Doing so brings some uncertainty to any of the results we reach. This uncertainty is what leads us to rely on some of the most important concepts of probability like expected values or prediction intervals. 2. Description 2.1 Relation Between Statistics and Probability In a way, probability lays the groundwork for stati..
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Bayes' Rule(Bayes' Theorem, Bayes' Law)Math/Probability 2019. 10. 14. 18:55
1. Overview In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer than can be done..