Consider a real-valued random process

where  and  is a positive integer. Here,  for  and 0 otherwise. The coefficients  are pairwise independent, zero-mean unit variance random variables.

Read the following statements about the random process and choose the correct option.

(i) The mean of the process  is independent of time .

(ii) The autocorrelation function  is independent of time  for all .

(Here,  is the expectation operation.)

The random variable  takes values in  with probabilities  and , where . Let  denote the entropy of  (in bits), parameterized by . Which of the following statements is/are TRUE?

A white Gaussian noise  with zero mean and power spectral density , when applied to a first-order RC low pass filter produces an output . At a particular time , the variance of the random variable  is _________.

Suppose  and  are independent and identically distributed random variables that are distributed uniformly in the interval . The probability that  is _________.

Let  be a random process, where amplitude  and phase  are independent of each other, and are uniformly distributed in the intervals  and , respectively.  is fed to an 8 -bit uniform mid-rise type quantizer. Given that the autocorrelation of  is , the signal to quantization noise ratio (in , rounded off to two decimal places) at the output of the quantizer is_________. 

A random variable , distributed normally as , undergoes the transformation , given in the figure. The form of the probability density function of  is (In the options given below,  are non-zero constants and  is piecewise continuous function)