Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2013

## Question

Part 1.

Consider $n$ independent flips of a coin having probability $p$ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $n=5$ and the sequence $HHTHT$ is observed, then there are 3 changeovers. Find the expected number of changeovers for $n$ flips. Hint: Express the number of changeovers as a sum of Bernoulli random variables.

Part 2.

Let $X_1,X_2,...$ be a sequence of jointly Gaussian random variables with covariance

$Cov(X_i,X_j) = \left\{ \begin{array}{ll} {\sigma}^2, & i=j\\ \rho{\sigma}^2, & |i-j|=1\\ 0, & otherwise \end{array} \right.$

Suppose we take 2 consecutive samples from this sequence to form a vector $X$, which is then linearly transformed to form a 2-dimensional random vector $Y=AX$. Find a matrix $A$ so that the components of $Y$ are independent random variables You must justify your answer.

Part 3.

Let $X$ be an exponential random variable with parameter $\lambda$, so that $f_X(x)=\lambda{exp}(-\lambda{x})u(x)$. Find the variance of $X$. You must show all of your work.

Part 4.

Consider a sequence of independent random variables $X_1,X_2,...$, where $X_n$ has pdf

\begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\ &+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align}.

Does this sequence converge in the mean-square sense? Hint: Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables $X_1,X_2,...$ converges in mean-square if and only if $E[|X_n-X_{n+m}|] \to 0$ as $n \to \infty$, for every $m>0$.