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).

Consider the random process
x(t) = U + Vt.
Where U is a zero mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean value of the random process at t = 2 is _________________

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

Two independent random variable X and Y are uniformly distributed in the interval [ - 1, 1]. The probability that max [X, Y] is less than 1/2 is