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| Suppose we want to know what the normal distribution function would look like in the form of a polynomial as the function is fairly ugly as it looks like this: |
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| Now the taylor series is slightly simpler. I would much rather take the derivative (or the antiderivative) of the first few terms of the taylor then the original function that is a for sure. Here is the first 8 terms of the taylor series, not that it is an even function. |
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| Now we want to find the approximate area under the curve in between the points -1.96 and 1.96, which is the 95% confidence level. This is the integral of the function: |
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| And this is the integral of the first 15 terms of the taylor series. |
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| Note how close the two values are. Now to totally believe in the taylor series, we must look at it graphically. Here is what happens when we increase the numbers of terms. Which is the blue line, which is compared to the graph of the normal curve in red. |
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| Taylor Series of Normal Distribution |
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