This document discusses moment generating functions (MGFs), which are defined as the expectation of e^tx, where x is a random variable and t is a real number for which the expectation is finite. The MGF completely determines the distribution of a random variable. Higher moments describe properties like symmetry, peakedness, and kurtosis of a probability distribution. The MGF can be used to find moments of a random variable.

1. 1.6 Moment Generating Functions

2. Moment Generating Functions (mgf) This is called the kth raw moment or kth moment about the origin. is the first moment about origin This implies that or first raw moment.

3. Is called the kth moment about the mean or the kth central moment. Therefore is called the second central moment. Moment Generating Functions

5. Higher moments are often used in statistics to give further descriptions of the probability distributions. Moment Generating Functions

6. Moment Generating Functions The third moment about the mean is used to describe the symmetry or skewness of a distribution. The fourth moment about mean is used to describe its “peakedness” or kurtosis. Kurtosis is a quantity indicative of the general form of a statistical frequency curve near the mean of the distribution.

7. Given below is the density of X Find Moment Generating Functions

8. Moment Generating Functions

9. Moment Generating Functions

10. Definition: Moment Generating Function Let X be a random variable with density f. The moment generating function of X (mgf) is denoted by and is given by Provided this expectation is finite for all real numbers in some open interval

11. Theorem: Let be the moment generating function for a random variable Then, Moment Generating Functions

12. Moment Generating Functions 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.