Chapter 3 - Lecture 4 Moments and Moment Generating Funct
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1 Chapter 3 - Lecture 4 and s October 7th, 2009 Chapter 3 - Lecture 4 and Moment Generating Funct
2 Central Skewness Chapter 3 - Lecture 4 and Moment Generating Funct
3 Central Skewness The expected value of a power of a random variable is called moment (or moment around 0). First moment or Expected value E(X ) Second moment E(X 2 ) Third moment E(X 3 ) and so on... Chapter 3 - Lecture 4 and Moment Generating Funct
4 Central Skewness Central Central moments are the expected value of the difference of a random variable and its expected value (or mean) to a power. It is also called moment about the mean. ((X E(X )) 2) Second central moment or variance E Third central moment E ( (X E(X )) 3) and so on... Chapter 3 - Lecture 4 and Moment Generating Funct
5 Central Skewness Introduction and central moments are useful because they give us useful information regarding the distribution of a random variable. Until now we know how to find the mean and the variance using moments and central moments. Another useful measure is skewness (see Chapter 1). Chapter 3 - Lecture 4 and Moment Generating Funct
6 Central Skewness Skewness measures the departure from symmetry Using the third and second central moments we can calculate skewness: E ((X E(X )) 3) σ 3 Chapter 3 - Lecture 4 and Moment Generating Funct
7 Central Skewness In State College there are families. I asked them how many kids they have and I get the following answers: 0 kids: 900 families 1 kid: 1700 families 2 kids: 4000 families 3 kids: 2100 families 4 kids: 700 families 5 kids: 400 families 6 kids: 200 families Find the skewness of X = number of kids in a family in SC. Chapter 3 - Lecture 4 and Moment Generating Funct
8 Moment generating Functions Is not always easy to calculate moments and central moments. Thats why we use moment generating functions. The moment generating function (mgf) of a discrete random variable X is defined to be ( ) M X (t) = E e tx = e tx p(x) x D Chapter 3 - Lecture 4 and Moment Generating Funct
9 Bernoulli random variable Find the moment generating function of a Bernoulli random variable Chapter 3 - Lecture 4 and Moment Generating Funct
10 Outline If the mgf exists and is the same for two distributions, then the two distributions are the same. In other words, the moment generating function uniquely specifies the probability distribution Chapter 3 - Lecture 4 and Moment Generating Funct
11 Outline In State College there are families. I asked them how many kids they have and I get the following answers: 0 kids: 900 families 1 kid: 1700 families 2 kids: 4000 families 3 kids: 2100 families 4 kids: 700 families 5 kids: 400 families 6 kids: 200 families Find the mgf of X = number of kids in a family in SC. Chapter 3 - Lecture 4 and Moment Generating Funct
12 If a variable Y has mgf the following find its distribution function: M Y (t) = e t +0.4e 2t e 3t e 4t e 5t e 6t Chapter 3 - Lecture 4 and Moment Generating Funct
13 Theorem Outline If the mgf exists then E(X r ) = M (r) X (0) Proof? Chapter 3 - Lecture 4 and Moment Generating Funct
14 Outline Lets go back to the example with the kids per family in State College. Use the Theorem in the previous page to find E(X ), E(X 2 ), E(X 3 ) and var(x ). Chapter 3 - Lecture 4 and Moment Generating Funct
15 If X has mgf M X (t) and Y = ax + b then M Y (t) = e bt M X (at) Proof? Chapter 3 - Lecture 4 and Moment Generating Funct
16 Section 3.4 page , 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57 Chapter 3 - Lecture 4 and Moment Generating Funct
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