# Variance of Integral of a real white Gaussian Noise Process

In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having infinite variance, so the resulting random variable also has infinite variance. This was a question asked in GATE exam conducted in India for Electronics and Communication stream

• But is it not an integration in time domain ? Dec 19 '20 at 13:50
• But is it not a time domain integration from t=5 to t =7 Dec 19 '20 at 13:55

Since $$W(t)$$ is assumed to be zero-mean, also the RV $$Y$$ is zero-mean. Hence, the variance of $$Y$$ is given by

\begin{align}\sigma_Y^2&=E\left\{Y^2\right\}\\&=E\left\{\int_{-\infty}^{\infty}W(t_1)\phi(t_1)dt_1\int_{-\infty}^{\infty}W(t_2)\phi(t_2)dt_2\right\}\\&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\phi(t_1)\phi(t_2)E\big\{W(t_1)W(t_2)\big\}dt_1dt_2\tag{1}\end{align}

where $$E\big\{W(t_1)W(t_2)\big\}$$ is the auto-correlation function $$R_W(t_2-t_1)$$ of $$W(t)$$. Now you just have to figure out the expression for $$R_W(\tau)$$ and solve the integral $$(1)$$.

• Isnt this integral also evaluates to infinity, because E[ W(t1) W(t2) ] is non zero only for t1 = t2 and E[W^2 (t) ] evaluates infinity , and the remaining is energy of phi (t) Dec 19 '20 at 14:37
• @Sreejith: No, you get a Dirac delta impulse inside the integral and you can use the sifting property to solve the integral. Dec 19 '20 at 15:23
• Its delta ( t1 - t2 ) and is zero except for t1 = t2, when t1 = t2 we get integral of delta (0). phi^2(t) ,which is different from integral of delta ( t ) phi ^2 ( t ). how do we apply sifting property for integral of delta (0) . phi^2 ( t) Dec 19 '20 at 15:39
• @Sreejith: You don't get an integral over $\delta(0)$, because $\delta(0)$ is meaningless. I guess you have to review the sifting property: $\int_{-\infty}^{\infty}\phi(t_2)\delta(t_2-t_1)dt_2=\ldots$. Dec 19 '20 at 15:47
• Thank you very much, I got it now Dec 19 '20 at 16:00