Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2016

## Question

Part 1.

Part 2.

A point $\omega$ is picked at random in the triangle shown below (all points are equally likely.) let the random variable $X(\omega)$ be the perpendicular distance from $\omega$ to be base as shown in the diagram.

(a) Find the cumulative distribution function (cdf) of $\mathbf{X}$.
(b) Find the probability distribution function (pdf) of $\mathbf{X}$.
(c) Find the mean of $\mathbf{X}$.
(d) What is the probability that $\mathbf{X}>h/3$.

Part 3.

Let $X$ and $Y$ be independent, jointly-distributed Poisson random variables with means with mean $\lambda$ and $\mu$. Let $Z$ be a new random variable defined as
$Z=X+Y$
(a) Find the probability mass function (pmf) of $\mathbf{Z}$.
(b) Show that the conditional probability mass function (pmf) of $X$ conditioned on the event ${Z=n}$ is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters $"n"$ and $"p"$) required to specify a binomial distribution $b(n,p)$).

Part 4.

Let $X(t)$ be a wide-sense stationary Gaussian random process with mean $\mu_x$ and autocorrelation function $R_xx(\tau)$. Let
$Y(t)=c_1X(t)-c_2X(t-T)$,
where $c_1,c_2$ and $T$ are real numbers. What is the probability that $Y(t)$ is less than or equal to a real number $/\gamma?$ Express your answer in terms of $c_1,c_2,\mu_x,\sigma_x^2$, and $R_xx(\tau), \gamma$ and the "phi function"
$\Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz$