1. Lecture 11 Pairs and Vector of Random Variables Last Time Pairs of R.Vs. Marginal PMF (Cont.) Joint PDF Marginal PDF Functions of Two R.Vs Expected Values Reading Assignment: Chapter 4.3 – 4.7 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_05_2008 11 - 1

2. Makeup Classes I will attend Networking 2009 in Aachen, Germany, and need to make-up the classes of 5/14 & 5/15 (3 hours) 4/30 17:30 – 18:20, 5/7 17:30 – 18:20, 5/8 8:10 – 9:00 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_05_2008 11 - 2

3. Lecture 11: Pair of R.V.s 5/7 Pairs of R.Vs. Functions of Two R.Vs Expected Values Conditional PDF Reading Assignment: Sections 4.6-4.9 5/8 Independence between Two R.Vs Bivariate R.V.s Random Vector Probability Models of N Random Variables Vector Notation Marginal Probability Functions Independence of R.Vs and Random Vectors Function of Random Vectors Reading Assignment: Sections 4.10-5.5 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_05_2008 11 - 3

4. Lecture 11: Pairs of R.Vs Next Time: Random Vectors Function of Random Vectors Expected Value Vector and Correlation Matrix Gaussian Random Vectors Sums of R. V.s Expected Values of Sums PDF of the Sum of Two R.V.s Moment Generating Functions Reading Assignment: Sections 5.1-6.3 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_04_2008 11 - 4

5. What have you learned about pair of R.Vs.? 16 - 5 Buffon's Needle Problem (AD: 1777): Throw a needle of length L at random on a floor covered by equi-distant parallel lines d units apart. What is the probability that the needle will cross at least one of the lines? (Note that in this case, L is not necessarily less than d.)

6. 11 - 6 If both L and d are known, Buffon's Needle experiment can be used to estimate the value of. Discussions the needle will intersect one of the lines if and only if

7. Buffon Needle Simulation http://www.ms.uky.edu/~mai/java/stat/buff.html 11 - 7

8. Alternative Way to Estimate the value of π? (-1,-1)(1,-1) (1,1)(-1,1) Let X, Y, be independent random variables uniformly distributed in the interval [-1,1] The probability that a point (X,Y) falls in the circle is given by SOLUTION  Generate N pairs of uniformly distributed random variates (u 1,u 2 ) in the interval [0,1).  Transform them to become uniform over the interval [-1,1), using (2u 1 -1,2u 2 -1).  Form the ratio of the number of points that fall in the circle over N Source: www.eng.ucy.ac.cy/christos/courses/ECE658/Lectures/RNG.ppt www.eng.ucy.ac.cy/christos/courses/ECE658/Lectures/RNG.ppt

9. Brain Teaser: Generating a Gaussian: Box-Muller method Generate Then are independent, Gaussian, zero mean, variance 1 You prove it!

35. 12 - 35

52. How about P[ X x| Y =y ] =? Example :