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# Geometric and Probabilistic Methods in Computer Science CS 530

UNM

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This 32 page Class Notes was uploaded by Trent Dare on Wednesday September 23, 2015. The Class Notes belongs to CS 530 at University of New Mexico taught by Staff in Fall. Since its upload, it has received 55 views. For similar materials see /class/212212/cs-530-university-of-new-mexico in ComputerScienence at University of New Mexico.

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Date Created: 09/23/15

Joe Kniss CS 530 Probabilistic and Geometric Methods Summing Random Variables m L ERF Plot 1 cdfx l erf Cdf There is an error in this equation what is it Adding 2 Random Vars Discrete Z XY X x1x2xn Yy1ay2ym Z x1 Y1ax1 Y2aquot39ax1 Yma X2 y1x2 ym xn y1 quotxn ym Adding Example How many ways can you get Z10 lt O O O O gt lt O O O O gt 1234 6789 X1234 Y6789 l lt O O O O O O O gt 7 8 910111213 X Y Z 78910111213 Adding 2 Random Vars But there is a Probability associated with each instance Xyj of X Y So what s the Prob of 2k PZ zk 213 xivyjlzkxiyj For any 2k add up the probabilities for all the ways Xyjzk Adding Example Assume X and Y are independent P xi9yj PxiPyj 7 8 910111213 PZ 10 PX1Py9 PX4Py6 4 0625 Adding as Convolution Zkxiyj xiZkyj Observe PltZ zkgt 210mm xi yjlzkxiyj So PltZ zkgt 2m m y 00 Note PXx0 if XX Convolution Example xizkyj flip and translate function 3 45 6quot 8 910111213 25 125 PZz 2PX7yPyy 7 8 9 10 11 1213 y Convolution Example xizkyj flip and translate function 3 4 5 6 quot7 8 9 10 111213 25 125 2PX8yPYy 7 8 9 10 11 1213 y Convolution Example xizkyj flip and translate function 3 4 5 6 7quot 8 9 10111213 2PX9yPYy 8 9 10 11 1213 Convolution Example xizkyj flip and translate function 3 4 5 6 78 9 10111213 25 u u u u u u u u u u u u 8 9 1O 11 12 13 EPX 10 ygtPYltygt y Discrete Convolution Given functions fX and gX f gx 2 f ygx y y OO Flip and translate gX Multiply elements of fand 9 Sum them up Subtraction of Random Vars ZXYjust like addition X x1x2xn Yy1ay2ym Z x1y1 x1y2quot x1ym x2 y1x2 ym xn ypmxn ym Subtraction Example How many ways can you get Z5 lt O O O O gt lt O O O O gt 1234 6789 X1234 Y6789 l lt gt 8 7 6 5 4 3 2 X Y Z 8 7 6 5 4 3 2 Subtracting 2 Random Vars But there is a Probability associated with each instance Xyj of X Y So what s the Prob of 2k PZ Zk 2130613371 xi ayj IZkxii yj For any 2k add up the probabilities for all the ways X yfzk Subtract Example Assume X and Y are independent P xi9yj PxiPyj 3 4 PZ 5 PX1Py6 PX 4 9 4 0625 Subtraction as Convolution Zkxiyj yjxizk Observe PltZ zkgt 210mm xi yjlzkxiyj So PZ zk 2Pxx zk x 00 Note PXx0 if XX Convolution Example yjxizk translate function no flip 25 125 PZz Convolution Example yjxizk translate function no flip 25 125 PZz 8 7 6 5 4 3 2 2PXxPYx 7 Convolution Example yjxizk translate function no flip PZ z Convolution Example yjxizk translate function no flip Difference as Disc Correlation Given functions fX and gx Convolve fX and gX f glt ltgtgtltxgt Efmgo x Negate argument to g y 00 Translate gx Sum up product of elements Meaning of Correlation How similar are PX and Py shifted by zk 8 7 6 5 4 3 2 Most similar when Py is shifted 5 Convolution Notes Convolution flips one of the arguments By convention second one Correlation doesn t flip Use convolution by negating second arg Sum rv Convolution when Independent Diff rv Correlation when Independent Continuous RV Sum Integration instead of summation me prYltsz sgtds Line integral through pXy where yzx Continuous Sum Example 190 zO5 Sum as Convolution When Xand Yare independent Way pxpy So pzz f 19X spyz Sds Which is a oonvolution of PX and Py pz Z px pYZ Difference of Cont RVs ZXY pZZ prYSaZ SdS Line integral along yX Z Continuous Difference Example my zO5 Sum as Correlation When Xand Yare independent Way pxpy 80 me prltsgtpYlts zgtds Which is a correlation of PX and Py 192Z PX PYZ Central Limit Theorem Sum a large number of iid rvs Identical Independently Distributed X1X2X3X Get a Gaussian X X 11m 1 N NM s a NO1x Central Limit Example Uniform Distribution

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