Why Doesn't My Calculation of c(0) Equal \(\sigma^2\)?

[ƒ(x)]

Homework Statement

Show that

c(0) = $\sigma$$^{2}$

Homework Equations

c($\ell$) = $\frac{1}{N-\ell}$ $\sum$ y$_{i}$y$_{i+\ell}$, with the summation going from i=1 to N-$\ell$

y(i) = x(i)-$\langle$x$\rangle$

$\langle$x$\rangle$=$\frac{1}{N}$$\sum$x(i)

$\langle$x²$\rangle$=$\frac{1}{N}$$\sum$x(i)²

Edit: the other summations go from 1 to N

Edit: forgot sigma

$\sigma$² = ($\langle$x²$\rangle$-$\langle$x$\rangle$)/(N-1)

The Attempt at a Solution

c(0) = $\frac{1}{N}$$\sum$y(i)y(i)
= $\frac{1}{N}$$\sum$(x(i)²-2x(i)$\langle$x$\rangle$+$\langle$x$\rangle$²)
= $\langle$x²$\rangle$-2$\langle$x$\rangle$²+$\langle$x$\rangle$²
= $\langle$x²$\rangle$-$\langle$x$\rangle$²
Where did I mess up?

 

[mighty2000]

It looks like you forgot to include the sigma in your equation for c(0). It should be c(0) = \sigma^{2} instead of c(0) = \langlex2\rangle-\langlex\rangle2. This is because \sigma^{2} is the variance, which is what c(0) represents. Also, in your second line, it should be y(i) instead of x(i). Other than those two errors, the rest of your solution looks correct.