Consider a real-valued random process

$$ f(t)=\sum\limits_{n=1}^N a_n p(t-n T), $$

where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and 0 otherwise. The coefficients $a_n$ 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 $f(t)$ is independent of time $t$.

(ii) The autocorrelation function $E[f(t) f(t+\tau)]$ is independent of time $t$ for all $\tau$. (Here, $E[\cdot]$ is the expectation operation.)

A source transmits a symbol $s$, taken from $\\{-4, 0, 4\\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r = s + w$, where the noise $w$ is zero mean with variance 4 and is independent of $s$.

Using $Q(x) = \frac{1}{\sqrt{2\pi}} \int\limits_{x}^{\infty} e^{-\frac{t^{2}}{2}} dt$, the optimum symbol error probability is _______.

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2,2]$ and $[0, 2\pi]$, respectively. $X(t)$ is fed to an 8-bit uniform mid-rise type quantizer.

Given that the autocorrelation of $X(t)$ is $R_X(\tau) = \frac{2}{3} \cos(2\pi f_0 \tau)$, the signal to quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is _________.

A random variable X, distributed normally as N(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of Y is

(In the options given below, a, b, c are non-zero constants and g(y) is piece-wise continuous function)