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).
| Symbols | a | b | c | d | e | f | g | h |
|---|---|---|---|---|---|---|---|---|
| Frequency of occurrence | $$\frac{1}{2}$$ | $${1 \over 4}$$ | $${1 \over 8}$$ | $${1 \over {16}}$$ | $${1 \over {32}}$$ | $${1 \over {64}}$$ | $${1 \over {128}}$$ | $${1 \over {128}}$$ |
The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as marked. The parameter $$\alpha$$ lies in the interval [0.25, 1]. The value of .. for which the capacity of this channel is maximized, is __________ (rounded off to two decimal places).
