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Variance of the sum of random independent variables

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    let [itex]x_{i}[/itex] be a random variable, and let [itex] y_{j} = \sum x_{i}[/itex].

    The variance of the random distribution of the [itex]x_{i}'s[/itex] is known, and each y is the sum of an equal amount of [itex]x_{i}'s[/itex], say N of them.

    How do I compute the variance of y in terms of [itex] \sigma^2_{x} [/itex] and N?

    2. Relevant equations

    [itex] \sigma^2_{y} = \sum\frac{(y - \mu_{y})^2}{M} [/itex]
     
  2. jcsd
  3. Mar 19, 2012 #2
    Also forgot to mention that I already know that [itex] \mu_{y} = N\mu_{x}[/itex].
     
  4. Mar 20, 2012 #3

    Ray Vickson

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    I don't understand your formula for [itex] \sigma^2_{y}[/itex], which would be false for every probability distribution I can think of. (It assumes Y is a uniformly-distributed random variable taking M distinct values.)

    Start with the simple case N=2: [itex]Y = X_1 + X_2,[/itex] where [itex] X_1,\; X_2[/itex] are independent. Once you have done that case, the general case follows almost immediately.

    RGV
     
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