# Variance of the sum of random independent variables

## Homework Statement

let $x_{i}$ be a random variable, and let $y_{j} = \sum x_{i}$.

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

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

## Homework Equations

$\sigma^2_{y} = \sum\frac{(y - \mu_{y})^2}{M}$

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Also forgot to mention that I already know that $\mu_{y} = N\mu_{x}$.

Ray Vickson
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## Homework Statement

let $x_{i}$ be a random variable, and let $y_{j} = \sum x_{i}$.

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

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

## Homework Equations

$\sigma^2_{y} = \sum\frac{(y - \mu_{y})^2}{M}$
I don't understand your formula for $\sigma^2_{y}$, 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: $Y = X_1 + X_2,$ where $X_1,\; X_2$ are independent. Once you have done that case, the general case follows almost immediately.

RGV