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# Moment Generating Functions

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Moment Generating Functions

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### Moment Generating Functions

1. 1. 1.6 Moment Generating Functions<br />
2. 2. Moment Generating Functions (mgf)<br />This is called the kth raw moment or kth moment <br />about the origin.<br /> is the first moment about origin <br />This implies that <br />or first raw moment.<br />
3. 3. Is called the kth moment about the mean <br />or the kth central moment.<br />Therefore<br />is called the second central moment. <br />Moment Generating Functions<br />
4. 4. <ul><li>The mean  is the first moment about origin and variance is the second moment about the mean.
5. 5. Higher moments are often used in statistics to give further descriptions of the probability distributions. </li></ul>Moment Generating Functions<br />
6. 6. Moment Generating Functions<br />The third moment about the mean is used to describe the symmetry or skewness of a distribution.<br />The fourth moment about mean is used to describe its “peakedness” or kurtosis.<br />Kurtosis is a quantity indicative of the general form of a statistical frequency curve near the mean of the distribution.<br />
7. 7. Given below is the density of X<br />Find <br />Moment Generating Functions<br />
8. 8. Moment Generating Functions<br />
9. 9. Moment Generating Functions<br />
10. 10. Definition: Moment Generating Function<br />Let X be a random variable with density f. The moment generating function of X (mgf) is denoted by <br />and is given by <br />Provided this expectation is finite for all real numbers <br /> in some open interval <br />
11. 11. Theorem: Let <br />be the moment generating function<br />for a random variable<br />Then,<br />Moment Generating Functions<br />
12. 12. Moment Generating Functions<br /> The moment generating function is unique and completely determines the distribution of the random variable; thus if two random variables have the same mgf, they have the same distribution (density). Proof of uniqueness of the mgf is based on the theory of transforms in analysis, and therefore we merely assert this uniqueness.<br />