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Math Help - Approximating Uniform Dist by CLT

  1. #1
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    Approximating Uniform Dist by CLT

    I'm given a question which asks the following:

    In adding n real numbers, each is rounded to the nearest integer. Assume that the round-off errors, X_{i}, i = 1,...,n, are independently distributed as U(-0.5,+0.5)

    Obtain the approximate distribution of the total error \sum X_{i} in the sum of the n numbers and hence find the probability that the absolute error in the sum is at most \frac{1}{2}\sqrt{n}.

    Now, I assume it wants me to approx to Normal dist by CLT, which is:

    <br />
\frac{\sqrt{n}(\overline{X} - \mu)}{\sigma} ~ N(0,1)

    From the Uniform dist, I know E(X) = 0, and Var(X) = \frac{1}{12}

    So does this mean that the approximate dist of \sum X_{i} is

    \overline{X_{i}}\sqrt{12n} ~ N(0,1)

    or am I missing something regarding the fact that it's the dist of the sum rather than just X itself?
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  2. #2
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    Quote Originally Posted by Zenter View Post
    I'm given a question which asks the following:

    In adding n real numbers, each is rounded to the nearest integer. Assume that the round-off errors, X_{i}, i = 1,...,n, are independently distributed as U(-0.5,+0.5)

    Obtain the approximate distribution of the total error \sum X_{i} in the sum of the n numbers and hence find the probability that the absolute error in the sum is at most \frac{1}{2}\sqrt{n}.

    Now, I assume it wants me to approx to Normal dist by CLT, which is:

    <br />
\frac{\sqrt{n}(\overline{X} - \mu)}{\sigma} ~ N(0,1)

    From the Uniform dist, I know E(X) = 0, and Var(X) = \frac{1}{12}

    So does this mean that the approximate dist of \sum X_{i} is

    \overline{X_{i}}\sqrt{12n} ~ N(0,1)

    or am I missing something regarding the fact that it's the dist of the sum rather than just X itself?
    Total error X=\sum X_i \sim N(0,n/12) (approx, by CLT)

    (Note: \overline{X_{i}}=0)

    CB
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  3. #3
    MHF Contributor matheagle's Avatar
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    \mu=0

    CB uses \overline{X} as the population mean

    The sample mean \overline{X} is not zero, it's a random variable and there is no subscript i to it.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    \mu=0

    CB uses \overline{X} as the population mean

    The sample mean \overline{X} is not zero, it's a random variable and there is no subscript i to it.
    There appears to be some part of the conversation missing here, my comment about \overline X_i is due to its appearence in the last expression in the original post, so the LHS of that expression is a numeric constant equal to zero and not a standard normal RV.

    X_i is the RV representing the i-th round off error and is \sim U(-0.5,0.5) so by definition \overline X_i =E( X_i)=0

    CB
    Last edited by CaptainBlack; August 25th 2009 at 11:04 PM.
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    Total error X=\sum X_i \sim N(0,n/12) (approx, by CLT)

    (Note: \overline{X_{i}}=0)

    CB
    I understand what you did here, but then using this for the part where I work out the probability, I get

    P(Z \leq \frac{\sqrt{12n}}{2})

    Which, I don't think would be the right answer, since I'm not given a numerical figure for n, meaning I can't work that ^ out.
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  6. #6
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    Quote Originally Posted by Zenter View Post
    I understand what you did here, but then using this for the part where I work out the probability, I get

    P(Z \leq \frac{\sqrt{12n}}{2})

    Which, I don't think would be the right answer, since I'm not given a numerical figure for n, meaning I can't work that ^ out.
    That's funny, 'cos I get \mathrm P(|Z|<\surd3). Any other offers?
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by halbard View Post
    That's funny, 'cos I get \mathrm P(|Z|<\surd3). Any other offers?
    Which is what I get as well, the \sqrt{n} terms all cancel.

    CB
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