Let $${X_1},\,{X_2},$$ and $${X_3}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,\,1} \right]$$. The probability $$P\left\{ {{X_1} + {X_2} \le {X_3}} \right\}$$ is ___________ .

The power spectral density of a real stationary random process X(t) is given by $$${S_x}\left( f \right) = \left\{ {\matrix{ {{1 \over W},\left| f \right| \le W} \cr {0,\left| f \right| > W} \cr } } \right.$$$

The value of the expectation $$$E\left[ {\pi X\left( t \right)X\left( {t - {1 \over {4W}}} \right)} \right]$$$
is ---------------.

Bits 1 and 0 are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below.

If the detection threshold is 1, the BER will be

Bits 1 and 0 are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below.

The optimum threshold to achieve minimum bit error rate (BER) is