Homework 1

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 1
Due Jan. 27, 2003

Reading

Wozencraft & Jacobs: Chapter 1 and Chapter 2 up to page 58.

Problems

1. Wozencraft & Jacobs: Problem 2.7
2. Wozencraft & Jacobs: Problem 2.11
3. Consider a random variable $X$ having a double-exponential (Laplacian) density,
   
   $$p_X(x) = a \mbox{e}^{-b \vert x\vert}, -\infty < x < \infty$$
   
   
   where $a$ and $b$ are positive constants.
   1. Determine the relationship between $a$ and $b$ such that $p_X(x)$ is a valid density function.
   2. Determine the corresponding probability distribution function $P_X(x)$.
   3. Find the probability that the random variable lies between 1 and 2.
   4. What is the probability that $X$ lies between 1 and 2 given that the magnitude of $X$ is less than 2.
4. Melvin Fooch, a former student in ECE 630, has found that the hours he spends working ($W$) and sleeping ($S$) in preparation for the quiz are random variables having the joint density
   
   $$p_{W,S}(w,s) = \left\{ \begin{array}{ll} K & \mbox{if $10 \... ..., $0 \leq s$} \\ 0 & \mbox{otherwise} \end{array} \right.$$
   
   
   What Melvin does not know, and even his best friends will not tell him, is that working only furthers his confusion and that his grade $G$ is given by
   
   $$G = 2.5(S-W) + 50.$$
   
   
   1. Determine the constant $K$.
   2. What is the probability that the maximum time he devotes to either working or sleeping is less than 10 hours?
   3. The instructor will pass poor Melvin if he achieves a grade of 75 or greater. What is the probability that this will occur?