The random variable

$$ Y=\int_{-\infty}^{\infty} W(t) \phi(t) d t, \quad \text { where } \phi(t)=\left\{\begin{array}{cc} 1, & 5 \leq t \leq 7 \\ 0, & \text { otherwise } \end{array}\right. $$

and $W(t)$ is a real white Gaussian noise process with two-sided power spectral density $S_W(f)=3 \mathrm{~W} / \mathrm{Hz}$, for all $f$. The variance of $Y$ is $\_\_\_\_$ .

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous of at least one of its bits is decoded wrongly. The probability that the received block is erroneous, is

A binary random variable $X$ takes the value +2 or -2 . The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is $\_\_\_\_$ .

In a digital communication system, a symbol $S$ randomly chosen from the set $\left\{s_1, s_2, s_3, s_4\right\}$ is transmitted. It is given that $s_1=-3, s_2=-1, s_3=+1$ and $s_4=+2$. The received symbol is $Y=S+W . W$ is a zero mean unit - variance Gaussian random variable and is independent of $S . P_i$ is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol $S=s_i$. The index $i$ for which the conditional symbol error probability $P_i$ is the highest is $\_\_\_\_$ .