Homework7 ECE302Fall2008sanghavi - Rhea

Instructions

Homework 7 can be downloaded here on the ECE 302 course website.

Problem 1: Arbitrary Random Variables

Let $$ F $$ be a non-decreasing function with

$\lim_{x\rightarrow -\infty} F(x) = 0 \mbox{ and } \lim_{x\rightarrow +\infty} F(x) = 1.$

Let $$ U $$ be a uniform random variable on [0,1].

(a) Let $X = F^{-1}(U)$ . What is the CDF of $$ X $$ ? (Note $F^{-1}$ is the inverse of $$ F $$ . A function $$ g $$ is the inverse of $$ F $$ if $$ F(g(x)) = x $$ for all $$ x $$ )
(b)How can you generate an exponential random variable from $$ U $$ ?

Michael Allen 7.1_ECE302Fall2008sanghavi
Suan-Aik Yeo 7.1_ECE302Fall2008sanghavi
Spencer Mitchell 7.1_ECE302Fall2008sanghavi
Hamad Al Shehhi_ECE302Fall2008sanghavi //comment to Spencer
Arie Lyles 7.1_ECE302Fall2008sanghavi //comment by Jared McNealis
Anand Gautam 7.1_ECE302Fall2008sanghavi
Monsu Mathew 7.1_ECE302Fall2008sanghavi

Problem 2: Gaussian Generation

The most popular random number generator in the computer language C is drand48; a call

       X = drand48()

results in X being a uniform random variable on [0,1]. How can you generate a gaussian random variable in C using drand48 ? (Hint: use 1(b) above, and problem 4 of HW 6. Consider generating a variable $$ D $$ as in problem 4 of HW6, along with another variable and relating these two to the Gaussian $$ X $$ or $$ Y $$ defined in that problem.)

Problem 3: A Random Parameter

A coin machine spits out a coin with a random bias $$ Q $$ . $$ Q=q $$ means that the probability of heads for that coin is $$ q $$ . The PDF of $$ Q $$ is $$ f_Q(q) = 2q $$ for $0 \leq q \leq 1$ . Jack tosses the coin once, and it lands heads. He then tosses the coin again. What is the probability that it will land heads again the second time, given that it landed heads the first time?

Problem 4: Debate Date

Hillary and Barack are to have a date. The time of arrival of each person is an exponential random variable with parameter $\lambda$ , and the two variables ( $$ H $$ and $$ B $$ for Hillary and Barack, respectively) are independent. What is the PDF of the time between their two arrivals? (Note: this time is always positive) (Hint: it is easier to first find the PDF of $$ Z = B - H $$ .)