Lecture 11 Pairs and Vector of Random Variables

Lecture 11 Pairs and Vector of Random Variables Last Time Pairs of R. Vs. n Marginal PMF (Cont. ) n Joint PDF n Marginal PDF n Functions of Two R. Vs n Expected Values Reading Assignment: Chapter 4. 3 – 4. 7 Probability & Stochastic Processes Yates & Goodman (2 nd Edition) 11 - NTUEE SCC_05_2008

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

Lecture 11: Pair of R. V. s 5/7 n Pairs of R. Vs. n Functions of Two R. Vs n Expected Values Conditional PDF Reading Assignment: Sections 4. 6 -4. 9 n n 5/8 n Independence between Two R. Vs Bivariate R. V. s Random Vector n Probability Models of N Random Variables n n n Vector Notation n Marginal Probability Functions n Independence of R. Vs and Random Vectors Function of Random Vectors Reading Assignment: Sections 4. 10 -5. 5 n Probability & Stochastic Processes Yates & Goodman (2 nd Edition) 11 - 3 NTUEE SCC_05_2008

Lecture 11: Pairs of R. Vs Next Time: n Random Vectors n n Function of Random Vectors n Expected Value Vector and Correlation Matrix n Gaussian Random Vectors Sums of R. V. s n Expected Values of Sums n PDF of the Sum of Two R. V. s n Moment Generating Functions Reading Assignment: Sections 5. 1 -6. 3 Probability & Stochastic Processes Yates & Goodman (2 nd Edition) 11 - 4 NTUEE SCC_04_2008

What have you learned about pair of R. Vs. ? 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. ) 16 - 5

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

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

Alternative Way to Estimate the value of π? (-1, 1) (1, 1) n n (-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 (2 u 1 -1, 2 u 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/ECE 658/Lectures/RNG. ppt n

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

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