GATE 2020 EC – Question 34
The random variable
$$Y=\int_{-\infty}^{\infty}W(t)\phi(t)\,dt,\quad\text{where }\phi(t)=\begin{cases}1;&5\le t\le7\\0;&\text{otherwise}\end{cases}$$
and $W(t)$ is a real white Gaussian noise process with two-sided power spectral density $S_W(f)=3$ W/Hz, for all $f$. The variance of $Y$ is ________.
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Correct answer: 6
Explanation
White noise with two-sided PSD 3 has autocorrelation $3\delta(\tau)$, so $\text{Var}(Y)=3\int\phi^2(t)\,dt=3\times2=6$.