Lecture 11

Pairs and Vector of Random Variables

Last TimePairs 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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_200811 - 1

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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_2008

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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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_2008

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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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_04_2008

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What have you learned about pair of R.Vs.?

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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.)

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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

Buffon Needle Simulation

http://www.ms.uky.edu/~mai/java/stat/buff.html

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Alternative Way to Estimate the value of π?

(-1,-1) (1,-1)

(1,1)(-1,1)

Area of Circle

Area of square 4

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 2 2Pr 1

4X Y

SOLUTION Generate N pairs of uniformly distributed random variates

(u1,u2) in the interval [0,1). Transform them to become uniform over the interval [-

1,1), using (2u1-1,2u2-1). Form the ratio of the number of points that fall in the

circle over NSource: www.eng.ucy.ac.cy/christos/courses/ECE658/Lectures/RNG.ppt

Brain Teaser: Generating a Gaussian: Box-Muller method Generate

Then

are independent, Gaussian, zero mean, variance 1

You prove it!

uniform and 21 xx

)2cos(xln2 211 xy

)2sin(ln2 212 xxy

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

Example:

otherwise.

,0

;20,2),(,

xyyxf YX