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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.

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  • $\begingroup$ is the e(t) in the definition of g(T) the n(t) you introduced before? $\endgroup$ – tibL Apr 19 '12 at 17:20
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    $\begingroup$ You need the autocorrelation function of the process, not just the variance. $\endgroup$ – Dilip Sarwate Apr 19 '12 at 17:25
  • $\begingroup$ Yes, the noise is white and e(t)=n(t). The text is now correct. $\endgroup$ – Arrigo Benedetti Apr 19 '12 at 21:57
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    $\begingroup$ 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. $\endgroup$ – Dilip Sarwate Apr 20 '12 at 11:06

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