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Math Help - Simulation

  1. #1
    Senior Member Sampras's Avatar
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    Simulation

    Suppose you have some random vector  \bold{X} = (X_1, \dots, X_n) with a density function  f(x_1, \dots, x_n) . Suppose you want to compute the expected value of some function of the random vector (e.g.  E[g(\bold{X})] ). We know that

     E[g(\bold{X})] = \int \int  \cdots \int g(x_1, \dots, x_n) f(x_1, \dots, x_n) dx_1 \cdots dx_n


    What is a good way of developing approximation methods for computing  E[g(\bold{X})] ? If  g(\bold{X}) = \bold{X} then we are just computing the expected value of  \bold{X} . So then we can reduce this question (in this case) to what are some good ways of approximating the expected value of a random vector?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Sampras View Post
    Suppose you have some random vector  \bold{X} = (X_1, \dots, X_n) with a density function  f(x_1, \dots, x_n) . Suppose you want to compute the expected value of some function of the random vector (e.g.  E[g(\bold{X})] ). We know that

     E[g(\bold{X})] = \int \int  \cdots \int g(x_1, \dots, x_n) f(x_1, \dots, x_n) dx_1 \cdots dx_n


    What is a good way of developing approximation methods for computing  E[g(\bold{X})] ? If  g(\bold{X}) = \bold{X} then we are just computing the expected value of  \bold{X} . So then we can reduce this question (in this case) to what are some good ways of approximating the expected value of a random vector?

    If you can sample from the distribution with density  f(x_1, \dots, x_n) then the mean of the function values for the sample is an unbiased estimator of the expectation of g(\bold{X}).

    CB
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  3. #3
    Senior Member Sampras's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    If you can sample from the distribution with density  f(x_1, \dots, x_n) then the mean of the function values for the sample is an unbiased estimator of the expectation of g(\bold{X}).

    CB
    In other words,  \lim\limits_{n \to \infty} \frac{g(\bold{X}^{(1)}) + \cdots + g(\bold{X}^{(n)})}{n} = E[g(\bold{X})] ?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Sampras View Post
    In other words,  \lim\limits_{n \to \infty} \frac{g(\bold{X}^{(1)}) + \cdots + g(\bold{X}^{(n)})}{n} = E[g(\bold{X})] ?
    Yes

    CB
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