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Thread: Simulation

  1. #1
    Senior Member Sampras's Avatar
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    Simulation

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

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

    $\displaystyle 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 $\displaystyle E[g(\bold{X})] $? If $\displaystyle g(\bold{X}) = \bold{X} $ then we are just computing the expected value of $\displaystyle \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 $\displaystyle f(x_1, \dots, x_n) $ then the mean of the function values for the sample is an unbiased estimator of the expectation of $\displaystyle g(\bold{X})$.

    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 $\displaystyle f(x_1, \dots, x_n) $ then the mean of the function values for the sample is an unbiased estimator of the expectation of $\displaystyle g(\bold{X})$.

    CB
    In other words, $\displaystyle \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, $\displaystyle \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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