Mean of the square of a sum of exponential terms

[grepecs]

Homework Statement

[/B]
Calculate \widehat{Y^{2}}

(i.e., the mean of the square of Y.

Homework Equations

Y=\sum_{k=0}^{N-1}y_{k}

where

y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k}

and

t=N\tau

The quantities y_{k} (or G_{k}) are statistically independent.

The Attempt at a Solution

\widehat{Y^{2}}=\widehat{G^{2}}e^{-2\gamma t}\sum_{k=0}^{N-1}e^{2\gamma \tau k}=\widehat{G^{2}}e^{-2\gamma t}(\ \frac{1-e^{2\gamma t}}{1-e^{2\gamma \tau}} )

However, the correct answer should be

\widehat{Y^{2}}=\widehat{G^{2}}\frac{1}{2\gamma\tau} (\ 1-e^{-2\gamma \tau} )

so it seems like I'm doing something wrong, or is it possible to somehow simplify my answer in order to get the correct one?

 

[Ray Vickson]

Homework Statement

[/B]
Calculate \widehat{Y^{2}}

(i.e., the mean of the square of Y.

Homework Equations

Y=\sum_{k=0}^{N-1}y_{k}

where

y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k}

and

t=N\tau

The quantities y_{k} (or G_{k}) are statistically independent.

The Attempt at a Solution

\widehat{Y^{2}}=\widehat{G^{2}}e^{-2\gamma t}\sum_{k=0}^{N-1}e^{2\gamma \tau k}=\widehat{G^{2}}e^{-2\gamma t}(\ \frac{1-e^{2\gamma t}}{1-e^{2\gamma \tau}} )

However, the correct answer should be

\widehat{Y^{2}}=\widehat{G^{2}}\frac{1}{2\gamma\tau} (\ 1-e^{-2\gamma \tau} )

so it seems like I'm doing something wrong, or is it possible to somehow simplify my answer in order to get the correct one?

Unless the quantities $G_k$ have mean zero ($E G_k = 0$) and are identically distributed for different $k$ (or, at least, all have the same second moments), the expression you give is incorrect; and in that case the one given in the book (or wherever) is also incorrect. As far as I can see, the only way you could obtain something like the second expression would be to have non-zero values of $E G_j G_k$ of the form $E G_j G_k = E(G^2) c_{jk}$, where $E(G^2) \equiv E(G_1^2) = E(G_2^2) = \cdots = E(G_{N-1}^2)$. This is because we have $Y = \sum_{k=1}^{N-1} w_k G_k$ (with $w_k = e^{-\gamma t} e^{\gamma \tau k}$), so that
Y^2 = \sum_{k=1}^{N-1} w_k^2 G_k^2 + 2 \sum_{1 \leq j < k \leq N-1} w_j w_k G_j G_k,
so you would need an appropriate value for the expectation of the second summation above, and that would depend on the form of the coefficients $c_{jk}$.