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Math Help - Difference of Random Variables

  1. #1
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    Difference of Random Variables

    In a solution I am trying to understand, we are given:
    A \sim N(\mu, 0.04)
    B \sim N(\mu, 0.04)

    And then the next step is written as:
    (\bar{A}-\bar{B}) \sim N(0, 0.08)

    Can someone please explain to me how this step is made? I can see why the mean would be zero, but why are the variances summed? Thank you.
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  2. #2
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    Quote Originally Posted by doopokko View Post
    In a solution I am trying to understand, we are given:
    A \sim N(\mu, 0.04)
    B \sim N(\mu, 0.04)

    And then the next step is written as:
    (\bar{A}-\bar{B}) \sim N(0, 0.08)

    Can someone please explain to me how this step is made? I can see why the mean would be zero, but why are the variances summed? Thank you.
    I assume A and B are independent.

    So you want a difference of two normal random variables to have a variance of zero. Which would mean no uncertainty at all in the value of A - B .......

    It can be proved that if U = a_1 X_1 + a_2 X_2 then:

    E(U) = a_1 E(X_1) + a_2 E(X_2)

    Var(U) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + 2 a_1 a_2 Cov(X_1, X_2).

    If X_1 and X_2 are independent then Cov(X_1, X_2) = 0.


    Property 2 here: http://en.wikipedia.org/wiki/Normal_distribution

    is of related interest.
    Last edited by mr fantastic; October 21st 2008 at 02:44 AM.
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