As shown in Figure (b). The output of the low pass filter is $$y(t)$$.
![GATE ECE 2017 Set 1 Communications - Random Signals and Noise Question 28 English 1](https://imagex.cdn.examgoal.net/RpHMgH488BS0NUhW5/p2vbfffe0gms4DFaCxvxR6siCgo04/6d1BSNuEfGPa58udtuHzSy/uploadfile.jpg)
![GATE ECE 2017 Set 1 Communications - Random Signals and Noise Question 28 English 2](https://gateclass.cdn.examgoal.net/cpUV6fKlYc9mpCMLg/QWO75ybMCNi0PbktCWGouTZ0570Ti/ujjFZwLGcGKjAvowZy7kQB/uploadfile.jpg)
Let $$E$$ be the expectation operator and consider the following statements :
$$\left( {\rm I} \right)$$ $$E\left( {X\left( t \right)} \right) = E\left( {Y\left( t \right)} \right)$$
$$\left( {{\rm I}{\rm I}} \right)$$ $$\,\,\,\,\,\,\,\,E\left( {{X^2}\left( t \right)} \right) = E\left( {{Y^2}\left( t \right)} \right)$$
$$\left( {{\rm I}{\rm I}{\rm I}} \right)\,$$ $$\,\,\,\,\,\,E\left( {{Y^2}\left( t \right)} \right) = 2$$
Select the correct option:
![GATE ECE 2016 Set 3 Communications - Random Signals and Noise Question 29 English](https://gateclass.cdn.examgoal.net/gHkrsq1vuwYuCANx7/lbXcrYIja03ltAlZSIr8BvbogyIWq/Uk0LmySG6oW8tmNIHcO9Tk/uploadfile.jpg)
$$x\left( t \right) = \sum\limits_{n = - \infty }^\infty {{\beta _n}g\left( {t - nT} \right),} $$ where $$g\left( t \right) = \left\{ {\matrix{ {1,} & {0 \le t \le T} \cr 0 & {otherwise} \cr } } \right.$$
If there is a null at $$f = {1 \over {3T}}$$ in the power spectral density of $$X(t)$$, then $$k$$ is _________.