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Basic Statistical analysis
The purpose:
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Before we can begin our journey in financial mathematics, we need to define some statistical quantities that will help us analyze financial data. The common descriptive statistics we will look at are mean, variance, covariance, probability distributions, normal distribution, and log normal distributions. We also need to define some commonly used finance terms that will be used throughout the "Current Developments" section of this website. We will look at examples of financial data and some statistical calculations. Nothing novel is presented in this section, however, it is important to first define terms.
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Financial Definitions:
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Example (Expectation Value):
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If I break a stick of unit length into three random pieces, what's the expected length of the largest piece?
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Note: In this example I assumed x and y (The points at which I break the stick) are independent. What If I
break the stick at point x and then at some later time, I break the longer piece at a random point y? Now the events are not independent. What is the expected length of the longest piece? I'll leave this problem to the reader. Let me know if you solve it (rdelgadillo0000@gmail.com).
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Example (Variance):
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Assume we have a point that's uniformly distributed on the surface of a ball centered at the origin in 3d space. The ball has a radius of R. What's the variance of it's x-coordinate?
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Important properties of the Normal distribution:
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Mathematical/Statistical Definitions:
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