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).

Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability (1 $$-$$ $$\in$$), and flipped with probability $$\in$$. For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error. For $$\in$$ = 0.1, the probability that a transmitted codeword is decoded correctly is __________ (rounded off to two decimal places).

Consider a channel over which either symbol x_A or symbol x_B is transmitted. Let the output of the channel Y be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for y given x_A and x_B are :

$${f_{\left. Y \right|{x_A}}}(y) = {e^{ - (y + 1)}}u(y + 1)$$,

$${f_{\left. Y \right|{x_B}}}(y) = {e^{(y - 1)}}(1 - u(y - 1))$$,

where, u( . ) is the standard unit step function. The probability of symbol error for this system is _________ (rounded off to two decimal places).

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, f(x), as shown in the figure.

Consider a 1 bit quantizer that maps positive samples to value $$\alpha$$ and others to value $$\beta$$. If $$\alpha$$* and $$\beta$$* are the respective choices for $$\alpha$$ and $$\beta$$ that minimize the mean square quantization error, then ($$\alpha$$* $$-$$ $$\beta$$*) = ___________ (rounded off to two decimal places).