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Lecture 10-2007

# Lecture 10-2007 - Lecture X Definition 2.3.3 Let X be a...

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Lecture X

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Definition 2.3.3. Let X be a random variable with cdf F X . The moment generating function (mgf) of X (or F X ), denoted M X (t) , is provided that the expectation exists for t in some neighborhood of 0. That is, there is an h >0 such that, for all t in – h < t < h , E [ e tX ] exists. ( 29 [ ] tX X e E t M =
If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist. More explicitly, the moment generating function can be defined as: ( 29 ( 29 ( 29 [ ] variables random discrete for and , variables random continuous for = = = - x tx X tx X x X P e t M dx x f e t M

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Theorem 2.3.2: If X has mgf M X ( t ), then where we define ( 29 ( 29 0 ) ( 0 = = t X n n n X t M dt d M ( 29 ( 29 0 n n X E X M =
First note that e tx can be approximated around zero using a Taylor series expansion: ( 29 ( 29 ( 29 ( 29 [ ] 2 3 0 0 2 0 3 0 2 3 2 3 1 1 0 0 0 2 6 1 2 6 tx t t t X M t E e E e te x t e x t e x t t E x t E x E x = = + - + - + - + = + + + + L L

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Note for any moment n: Thus, as t 0 ( 29 ( 29 1 2 2 n n n n n X X n d M M t E x E x t E x t dt + + = = + + + L ( 29 ( 29 [ ] n n X x E M = 0
Leibnitz’s Rule: If f ( x ,θ), a (θ), and b (θ) are differentiable with respect to θ, then ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 + - = θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ b a b a dx x f b d d a f a d d b f dx x f d d , , , ,

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Lecture 10-2007 - Lecture X Definition 2.3.3 Let X be a...

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