Let the relevant bandwidth $(B)$ of a digital communication system be 1 MHz and $k T=-174 \mathrm{dBm} / \mathrm{Hz}$, where $k$ is Boltzmann's constant and ' $T$ ' is equivalent noise temperature of the receiver. The power ( $S$ ) of signal received through an additive Gaussian channel is -80 dBm .

Which of the following options is/are TRUE about Shannon capacity ( $C$ ) of the channel?

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 frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of "yes/no" questions which are assumed to be truthfully answered. The average number of questions when asked in the most efficient sequence, to determine the chosen symbol, is _____________ (rounded off to two decimal places).