2
$\begingroup$

Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the variance of $g(T)$ defined as

$g(T)=\int_0^Tx(t)n(t)dt.$

Any references to elementary textbooks on stochastic processes are also welcome.

  • 1
    If the autocorrelation function is E[n(t)n(s)] = R_n(t-s)=\begin{cases}\sigma^2,&t=s,\\0,&t\neq s,\end{cases} then the integral expression in Nate Eldredge's answer gives $\operatorname{var}(g(T))=0$. If the autocorrelation function is $\sigma^2\delta(t-s)$ (note the difference) then see my comment on that answer as well as [this question](http://math.stackexchange.com/q/134193/15941).2012-04-20

0 Answers 0