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Derivation of the Normal Distribution

Derivation of the Normal Distribution. Lecture VI. Derivation of the Normal Distribution Function. The order of proof of the normal distribution function is to start with the standard normal:.

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Derivation of the Normal Distribution

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  1. Derivation of the Normal Distribution Lecture VI

  2. Derivation of the Normal Distribution Function • The order of proof of the normal distribution function is to start with the standard normal:

  3. First, we need to demonstrate that the distribution function does integrate to one over the entire sample space, which is - to . This is typically accomplished by proving the constant. • Let us start by assuming that

  4. Squaring this expression yields

  5. Polar Forms • Polar Integration: The notion of polar integration is basically one of a change in variables. Specifically, some integrals may be ill-posed in the traditional Cartesian plane, but easily solved in a polar space. • By polar space, any point (x,y) can be written in a trigonometric form:

  6. As an example, take Some of the results for this function are:

  7. Polar Values

  8. The workhorse in this proof is Greene’s theorem. We know from univariate calculus that • In multivariate space, this becomes

  9. Primary function

  10. Graph in X, Y Space

  11. Transformed to Polar Coordinates

  12. Back to the Normal

  13. General Normal Distribution • The expression above is the expression for the standard normal. A more general form of the normal distribution function can be derived by defining a transformation function.

  14. By the change in variable technique, we have

  15. Expected Values • Definition 2.2.1. The expected value or mean of a random variable g(X) noted by E[g(X)] is

  16. The most general form used in this definition allows for taking the expectation of the function g(X). Strictly speaking, the mean of the distribution is found where g(X)=X, or

  17. Example

  18. Theorem 2.2.1. Let X be a random variable and let a, b, and c be constants. Then for any functions g1(X) and g2(X) whose expectations exist: • E[ag1(X) + bg2(X) + c]=aE[g1(X)] + bE[g2(X)] + c. • If g1(X)  0 for all X, then E[g1(X)]  0. • If g1(X) g2(X) for all X, then E[g1(X)] E[g2(X)] • If ag1(X) b for all X, then aE[g1(X)] b

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