(→‎Problem 3: A Random Parameter)
 
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[[Category:ECE302Fall2008_ProfSanghavi]]
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[[Category:probabilities]]
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[[Category:ECE302]]
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[[Category:problem solving]]
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== Instructions ==
 
== Instructions ==
 
Homework 7 can be [https://engineering.purdue.edu/ece302/homeworks/HW7FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website].
 
Homework 7 can be [https://engineering.purdue.edu/ece302/homeworks/HW7FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website].
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*(a) Let <math>X = F^{-1}(U)</math>. What is the CDF of <math>X</math>?  (Note <math>F^{-1}</math> is the inverse of <math>F</math>. A function <math>g</math> is the inverse of <math>F</math> if <math>F(g(x)) = x</math> for all <math>x</math>)
 
*(a) Let <math>X = F^{-1}(U)</math>. What is the CDF of <math>X</math>?  (Note <math>F^{-1}</math> is the inverse of <math>F</math>. A function <math>g</math> is the inverse of <math>F</math> if <math>F(g(x)) = x</math> for all <math>x</math>)
 
*(b)How can you generate an exponential random variable from <math>U</math>?
 
*(b)How can you generate an exponential random variable from <math>U</math>?
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*[[Michael Allen 7.1_ECE302Fall2008sanghavi]]
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*[[Suan-Aik Yeo 7.1_ECE302Fall2008sanghavi]]
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*[[Spencer Mitchell 7.1_ECE302Fall2008sanghavi]]
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*[[Hamad Al Shehhi_ECE302Fall2008sanghavi]] //comment to Spencer
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*[[Arie Lyles 7.1_ECE302Fall2008sanghavi]] //comment by Jared McNealis
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*[[Anand Gautam 7.1_ECE302Fall2008sanghavi]]
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*[[Monsu Mathew 7.1_ECE302Fall2008sanghavi]]
  
 
== Problem 2: Gaussian Generation ==
 
== Problem 2: Gaussian Generation ==
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         X = drand48()
 
         X = drand48()
  
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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)
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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 <math>D</math> as in problem 4 of HW6, along with another variable and relating these two to the Gaussian <math>X</math> or <math>Y</math> defined in that problem.)
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*[[Nicholas Browdues_ECE302Fall2008sanghavi]]
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*[[Andrew Hermann_ECE302Fall2008sanghavi]] (reply/discussion: Brian Thomas)
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*[[Kristin Wing_ECE302Fall2008sanghavi]]
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*[[Anand Gautam_ECE302Fall2008sanghavi]]
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*[[Seraj Dosenbach_ECE302Fall2008sanghavi]]
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*[[Daniel Truax_ECE302Fall2008sanghavi]]
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*[[Emir Kavurmacioglu_ECE302Fall2008sanghavi]]
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*[[Chris Rush_ECE302Fall2008sanghavi]]
  
 
== Problem 3: A Random Parameter ==
 
== Problem 3: A Random Parameter ==
 
A coin machine spits out a coin with a random bias <math>Q</math>. <math>Q=q</math> means that the probability of heads for that coin is <math>q</math>. The PDF of <math>Q</math> is <math>f_Q(q) = 2q</math> for <math>0 \leq q \leq 1</math>. 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?
 
A coin machine spits out a coin with a random bias <math>Q</math>. <math>Q=q</math> means that the probability of heads for that coin is <math>q</math>. The PDF of <math>Q</math> is <math>f_Q(q) = 2q</math> for <math>0 \leq q \leq 1</math>. 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?
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*[[Suan-Aik Yeo 6.1_ECE302Fall2008sanghavi]]
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*[[Priyanka Savkar 7.3_ECE302Fall2008sanghavi]]
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*[[Justin Mauck 7.3_ECE302Fall2008sanghavi]]
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*[[Gregory Pajot 7.3_ECE302Fall2008sanghavi]]
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*[[Patrick M. Avery Jr. 7.3_ECE302Fall2008sanghavi]]
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*[[Divyanshu Kamboj 7.3_ECE302Fall2008sanghavi]]
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*[[Ken Pesyna 7.3_ECE302Fall2008sanghavi]]
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*[[Junzhe Geng 7.3_ECE302Fall2008sanghavi]]
  
 
== Problem 4: Debate Date ==
 
== Problem 4: Debate Date ==
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Hillary and Barack are to have a date. The time of arrival of each person is an exponential random variable with parameter <math>\lambda</math>, and the two variables (<math>H</math> and <math>B</math> 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 <math>Z = B - H</math>.)
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*[[7.4 Katie Pekkarinen_ECE302Fall2008sanghavi]]
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[[7.4 Joon Young Kim_ECE302Fall2008sanghavi]]
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*[[7.4 Spencer Mitchell  (answer to the // at the bottom of Joon Kim)_ECE302Fall2008sanghavi]]
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[[7.4 Ben Carter_ECE302Fall2008sanghavi]]
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[[7.4 Divyanshu Kamboj_ECE302Fall2008sanghavi]]
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*[[7.4 Chris Wacnik_ECE302Fall2008sanghavi]]
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[[7.4 Shao-Fu Shih_ECE302Fall2008sanghavi]]
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[[7.4 Steven Streeter_ECE302Fall2008sanghavi]]
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[[7.4 Josh Long_ECE302Fall2008sanghavi]]
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[[7.4 Joe Gutierrez_ECE302Fall2008sanghavi]]
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----
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[[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]]

Latest revision as of 11:57, 22 November 2011


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 $?

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 $.)


7.4 Joon Young Kim_ECE302Fall2008sanghavi

7.4 Ben Carter_ECE302Fall2008sanghavi

7.4 Divyanshu Kamboj_ECE302Fall2008sanghavi

7.4 Shao-Fu Shih_ECE302Fall2008sanghavi

7.4 Steven Streeter_ECE302Fall2008sanghavi

7.4 Josh Long_ECE302Fall2008sanghavi

7.4 Joe Gutierrez_ECE302Fall2008sanghavi

Back to ECE302 Fall 2008 Prof. Sanghavi

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang