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Math Help - Basic Statistics

  1. #1
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    Basic Statistics

    Hey,

    This is an easy problem, but it has been awhile since I have done basic statistics. If anyone can refresh my memory, that would be greatly appreciated. I believe the answer to the second problem is 100, as they are related by squares...but I am not 100% sure.



    Xi is the result of the ith run of a Monte Carlo simulation, i = 1, 2... 1000. Let the Xi be independent and identically distributed with mean 10-4 and variance 10-6.

    (a) What is the standard deviation of Y, where Y is the average of the Xi?

    (b) If we wish to reduce the standard deviation of Y by a factor of 10, how many additional simulation runs do we need to do, above and beyond the 1000 that we have already done?



    Thanks,
    Ryan
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  2. #2
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    Quote Originally Posted by Ryan J View Post
    Hey,

    This is an easy problem, but it has been awhile since I have done basic statistics. If anyone can refresh my memory, that would be greatly appreciated. I believe the answer to the second problem is 100, as they are related by squares...but I am not 100% sure.



    Xi is the result of the ith run of a Monte Carlo simulation, i = 1, 2... 1000. Let the Xi be independent and identically distributed with mean 10-4 and variance 10-6.

    (a) What is the standard deviation of Y, where Y is the average of the Xi?

    (b) If we wish to reduce the standard deviation of Y by a factor of 10, how many additional simulation runs do we need to do, above and beyond the 1000 that we have already done?



    Thanks,
    Ryan
    You can apply the Central Limit Theorem: Central limit theorem - Wikipedia, the free encyclopedia

    Also, read this: Sampling Distribution.
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  3. #3
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    I'm not sure how well that would work. Oh, and of course when it says 10-4, those are powers, meaning 10^-4. But anyway, I think I'll lose just the definition of SD, which states SD= Sqrt(E(x^2)-(E(x))^2)

    thanks though.

    -Ryan
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  4. #4
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    Quote Originally Posted by Ryan J View Post
    Hey,

    This is an easy problem, but it has been awhile since I have done basic statistics. If anyone can refresh my memory, that would be greatly appreciated. I believe the answer to the second problem is 100, as they are related by squares...but I am not 100% sure.



    Xi is the result of the ith run of a Monte Carlo simulation, i = 1, 2... 1000. Let the Xi be independent and identically distributed with mean 10-4 and variance 10-6.

    (a) What is the standard deviation of Y, where Y is the average of the Xi?

    (b) If we wish to reduce the standard deviation of Y by a factor of 10, how many additional simulation runs do we need to do, above and beyond the 1000 that we have already done?



    Thanks,
    Ryan
    For (b), you need n = 100,000 so another 99,000 runs are required:

    \sigma_Y = \frac{\sigma_X}{\sqrt{n}}.

    \sigma_{Y,\, \text{old}} = \frac{\sigma_X}{\sqrt{1000}}.

    \sigma_{Y,\, \text{new}} = \frac{\sigma_X}{10 \sqrt{1000}} = \frac{\sigma_X}{\sqrt{100000}} \Rightarrow n = 100,000.
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  5. #5
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    Quote Originally Posted by Ryan J View Post
    I'm not sure how well that would work. [snip] But anyway, I think I'll lose just the definition of SD, which states SD= Sqrt(E(x^2)-(E(x))^2)

    thanks though.

    -Ryan
    ??? !!!

    The Central Limit Theorem is required. Read the second link carefully.

    By the way, I thought I'd just mention that there's no need to know what the distribution of Xi is (why?).
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  6. #6
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    great thinking; I was not even thinking about that, but you are right. There is a couple extra points for my last assignment.

    Thanks again.

    -Ryan
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  7. #7
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    did not even notice second link, which is the more statistically accurate way of approaching the problem. And yes, almost every distribution becomes normal when you average them...however there are a few rare exceptions, that will never normalize.

    -Ryan
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