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Chapter 3c

# Chapter 3c - There are two questions that we avoided in our...

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There are two questions that we avoided in our discussions of the special discrete distributions: (1) How did we get from E(X) = Σ x*p(x) and V(X) = E(x 2 )- [E(x)] 2 to the formulas specified for the means and variances for the binomial, geometric, ….. distributions? Note: Since x isn’t always bounded, it is difficult to use the above formulas for E(x) and E(x 2 ). The answer to (1) is to use the moment generating function to get the moments. What are moments? The mean of a distribution is the 1 st moment. The 2 nd moment is E(x 2 ). How do we find the moment generating function? The moment generating function is denoted m x (t) and calculated as E(e tx ). It is unique for each distribution. For discrete probability distributions, this means m x (t) = Σ e tx *f(x). Evaluating these sums requires the application of special theorems and sums. How do we get the moments from the moment generating function? Find Binomial Distribution: denote q=1-p m x (t) = E(e tx ) = Σ e tx *f(x) = = (q +pe t ) n This requires use of the binomial theorem: (a+b) n =

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Chapter 3c - There are two questions that we avoided in our...

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