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Let $X(t)$ be a wide sense stationary random process with the power spectral density $S_{X}(f)$ as shown in Figure (a), where $f$ is in Hertz$(Hz)$. The random process $X(t)$ is input to an ideal lowpass filter with the frequency response

$$H(f)  = \begin{cases} 1, & \mid f \mid \leq \frac{1}{2} Hz \\ 0, &\mid f \mid > \frac{1}{2} Hz \end{cases}$$

as shown in Figure(b).The output of the lowpass filter is $Y(t)$.

Let $E$ be the expectation operator and consider the following statement:

  1. $E(X(t))=E(Y(t))$
  2. $E(X^{2}(t))=E(Y^{2}(t))$
  3. $E(Y^{2}(t))=2$

Select the correct option:

  1. only I is true
  2. only II and III are true
  3. only I and II are true
  4. only I and III are true
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