- #1
grepecs
- 17
- 0
Homework Statement
[/B]
Calculate [itex]\widehat{Y^{2}}[/itex]
(i.e., the mean of the square of [itex]Y[/itex].
Homework Equations
[tex]Y=\sum_{k=0}^{N-1}y_{k}[/tex]
where
[tex]y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k}[/tex]
and
[tex]t=N\tau[/tex]
The quantities [itex]y_{k}[/itex] (or [itex]G_{k}[/itex]) are statistically independent.
The Attempt at a Solution
[tex]\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}} ) [/tex]
However, the correct answer should be
[tex]\widehat{Y^{2}}=\widehat{G^{2}}\frac{1}{2\gamma\tau} (\ 1-e^{-2\gamma \tau} )[/tex]
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?