Variance of the sum of random independent variables

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Homework Statement



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?

Homework Equations



[itex] \sigma^2_{y} = \sum\frac{(y - \mu_{y})^2}{M} [/itex]
 

Answers and Replies

  • #2
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Also forgot to mention that I already know that [itex] \mu_{y} = N\mu_{x}[/itex].
 
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Ray Vickson
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Homework Statement



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?

Homework Equations



[itex] \sigma^2_{y} = \sum\frac{(y - \mu_{y})^2}{M} [/itex]
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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