Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2015

## Question

Part 1.

If $X$ and $Y$ are independent Poisson random variables with respective parameters $\lambda_1$ and $\lambda_2$, calculate the conditional probability mass function of $X$ given that $X+Y=n$.

Part 2.

Let $Z(t), t\ge 0$, be a random process obtained by switching between the values 0 and 1 according to the event times in a counting process $N(t)$. Let $P(Z(0)=0)=p$ and

$P(N(t)=k) = \frac{1}{1+\lambda t}(\frac{\lambda t}{1+\lambda t})^k$

for $k = 0, 1, ...$. Find the pmf of $Z(t)$.

Part 3.

Let $X$ and $Y$ be independent identically distributed exponential random variables with mean $\mu$. Find the characteristic function of $X+Y$.

Consider a sequence of independent and identically distributed random variables $X_1,X_2,... X_n$, where each $X_i$ has mean $\mu = 0$ and variance $\sigma^2$. Show that for every $i=1,...,n$ the random variables $S_n$ and $X_i-S_n$, where $S_n=\sum_{j=1}^{n}X_j$ is the sample mean, are uncorrelated.