1. Problem: Find the variance of $3X + 4Y$ where $X$ and $Y$ are independent random variables with variances 2 and 3 respectively. Formula: For independent variables, $\mathrm{Var}(aX + bY) = a^2 \mathrm{Var}(X) + b^2 \mathrm{Var}(Y)$. Calculation: $$\mathrm{Var}(3X + 4Y) = 3^2 \times 2 + 4^2 \times 3 = 9 \times 2 + 16 \times 3 = 18 + 48 = 66$$ 2. Problem: Find the range of the correlation coefficient. Answer: The correlation coefficient $\rho$ satisfies $-1 \leq \rho \leq 1$. 3. Problem: Define SSS process. Answer: A Strict Sense Stationary (SSS) process is a stochastic process whose joint probability distribution does not change when shifted in time. This means all statistical properties are invariant under time shifts. 4. Problem: Define steady state distribution of Markov Chain. Answer: The steady state distribution is a probability distribution $\pi$ satisfying $\pi P = \pi$, where $P$ is the transition matrix. It represents the long-term behavior of the Markov chain. 5. Problem: Write any two applications of a Bernoulli process. Answer: - Modeling success/failure trials like coin tosses. - Packet arrival in communication networks. 6. Problem: State any two properties of Poisson Process. Answer: - The number of events in disjoint intervals are independent. - The number of events in an interval of length $t$ follows a Poisson distribution with parameter $\lambda t$. 7. Problem: Compute the mean value of the random process $\{X(t)\}$ whose autocorrelation function is $R_{xx}(\tau) = 25 + \frac{4}{1 + 6\tau^2}$. Explanation: For a wide-sense stationary process, $R_{xx}(0) = E[X(t)^2] = \mathrm{Var}(X) + (E[X])^2$. Since $R_{xx}(\tau)$ has a constant term 25, which is $(E[X])^2$, the mean is: $$E[X] = \sqrt{25} = 5$$ 8. Problem: Given $R_{xx}(\tau) = 25 + \frac{4}{1 + 6\tau^2}$ for a stationary process with no periodic components, find mean and variance. Mean: $$E[X] = \sqrt{25} = 5$$ Variance: $$\mathrm{Var}(X) = R_{xx}(0) - (E[X])^2 = \left(25 + \frac{4}{1 + 0}\right) - 25 = 25 + 4 - 25 = 4$$ 9. Problem: Define a system and when is it called linear. Answer: A system is a set of rules that maps input signals to output signals. It is called linear if it satisfies the principles of superposition: additivity and homogeneity. 10. Problem: Given system function $$h(t) = \begin{cases} \frac{1}{2c} & |t| \leq c \\ 0 & |t| > c \end{cases}$$ Find the relationship between power spectrum of input and output processes. Answer: The power spectrum of output $S_y(\omega)$ relates to input $S_x(\omega)$ by $$S_y(\omega) = |H(\omega)|^2 S_x(\omega)$$ where $H(\omega)$ is the Fourier transform of $h(t)$. For this $h(t)$, $$H(\omega) = \int_{-c}^c \frac{1}{2c} e^{-j\omega t} dt = \frac{\sin(\omega c)}{\omega c}$$ Thus, $$S_y(\omega) = \left| \frac{\sin(\omega c)}{\omega c} \right|^2 S_x(\omega)$$