Consider a real-valued random process
$$ f(t)=\sum\limits_{n=1}^N a_n p(t-n T), $$
where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and 0 otherwise. The coefficients $a_n$ are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the correct option.
(i) The mean of the process $f(t)$ is independent of time $t$.
(ii) The autocorrelation function $E[f(t) f(t+\tau)]$ is independent of time $t$ for all $\tau$. (Here, $E[\cdot]$ is the expectation operation.)
The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<\frac{1}{2}$.
Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?
$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by:
$$ \begin{aligned} & P(X=0, Y=0)=0.06 \\ & P(X=0, Y=1)=0.14 \\ & P(X=1, Y=0)=0.24 \\ & P(X=1, Y=1)=0.56 \end{aligned} $$
The mutual information $I(X ; Y)$ is ___________(rounded off to two decimal places).
The Nyquist plot of a system is given in the figure below. Let $\omega_{\mathrm{P}}, \omega_Q, \omega_R$, and $\omega_{\mathrm{S}}$ be the positive frequencies at the points $P, Q, R$, and $S$, respectively. Which one of the following statements is TRUE?
