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Math Help - Generating functions...need some help here

  1. #1
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    Generating functions...need some help here

    A textbook defines the generating function f of a random variable X as:

    f(\theta) = E[\theta^X] = \sum_{k} \theta^kP(X=k)

    Can anybody please explain that to me?

    So far I know:
    Generating function of random variable X can also be defined as:
    X(\theta) = x_0 + x_1 \theta + x_2 \theta^2 + x_3 \theta^3 + ...

    So if we do differentiation:
    X'(\theta) = x_1  + 2x_2 \theta + 3x_3 \theta^2 + ... = \sum_{k} kx_k \theta^{k-1}

    Taking \theta = 1,
    X'(1) = \sum_{k} kx_k = E[X]

    How is this related to the very first equation?

    Any help is much appreciated.
    Last edited by chopet; January 31st 2008 at 07:30 AM.
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  2. #2
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    Quote Originally Posted by chopet View Post
    A textbook defines the generating function f of a random variable X as:

    f(\theta) = E[\theta^X] = \sum_{k} \theta^kP(X=k)
    What is your question? In this scenario, the generating function is just the expectation of the probability mass function.
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  3. #3
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    ok. I am confused. I thought the equation was meant to be a generality of all cases, not just the case for "expectation of probability mass".
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  4. #4
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    Quote Originally Posted by chopet View Post
    A textbook defines the generating function f of a random variable X as:

    f(\theta) = E[\theta^X] = \sum_{k} \theta^kP(X=k)

    Can anybody please explain that to me?

    So far I know:
    Generating function of random variable X can also be defined as:
    X(\theta) = x_0 + x_1 \theta + x_2 \theta^2 + x_3 \theta^3 + ...
    Ask yourself: "What are the x_i's here?"

    RonL
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  5. #5
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    Lemme try to reconstruct my understanding after your inputs.

    X is random variable 0,1,2,3,... with corresponding probability densities x_0, x_1, x_2, x_3... (which all add to 1).

    We define the generating function of X as:
    X(\theta) = x_0 + x_1 \theta + x_2 \theta^2 + ...

    Turning it around, we can now see a new random variable \theta^X (this is the key part which i couldn't see before), and the generating function is also the Expectation of this new variable, E(\theta^X)

    Of course, this is not to be confused with E(X), which is derived from the differentiation of the 1st equation, and equating \theta = 1.


    Am I correct?
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