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# lect10 - EE 278 Lecture Notes 10 Course Summary Basic...

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EE 278 Lecture Notes 10 Course Summary Basic Probability and Random Variables First and Second Moments, Correlation Gaussians Convergence and Limit Theorems Classes of Random Processes LTIS with WSS Process Input Linear Estimation EE 278: Course Summary Page 10 – 1

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Basic Probability and Random Variables Sample space Ω , outcome ω , events 2 Ω , probability measure P( · ) Law of total probability: events: P( B ) = i P( A i B ) if A i ’s partition Ω pmf: p X ( x ) = y p ( x,y ) pdf: f X ( x ) = i f X,Y ( x,y ) dy mixed: f Y ( y ) = θ p Θ ( θ ) f Y | Θ ( y | θ ) p Θ ( θ ) = i f Y ( y ) p Θ | Y ( θ | y ) dy Bayes rule events: P( A j | B ) = P( B | A j ) P i P( B | A i )P( A i ) P( A j ) if A i ’s partition Ω pmf: p X | Y ( x | y ) = p Y | X ( y | x ) P x p Y | X ( y | x ) p X ( x ) p X ( x ) EE 278: Course Summary Page 10 – 2
pdf: f X | Y ( x | y ) = f Y | X ( y | x ) R f X ( x ) f Y | X ( y | x ) dx f X ( x ) mixed: p Θ | Y ( θ | y ) = f Y | Θ ( y | θ ) P θ p Θ ( θ ) f Y | Θ ( y | θ ) p Θ ( θ ) f Y | Θ ( y | θ ) = p Θ | Y ( θ | y ) R f Y ( y ) p Θ | Y ( θ | y ) dy f Y ( y ) Review : HW3 Extra Problem #3 Signal detection: MAP, ML, and minimum distance decoding rules Review : HW3 #3,4, Sample Midterm #6, Midterm #4 EE 278: Course Summary Page 10 – 3

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Independence events: for all subsets of the set of events A 1 ,A 2 ,...,A n , P( A i 1 ,A i 2 ,...,A i k ) = p k j =1 P( A i j ) cdf: F X ( x ) = p n i =1 F X i ( x i ) for all x pmf: p X ( x ) = p n i =1 p X i ( x i ) for all x pdf: f X ( x ) = p n i =1 f X i ( x i ) for all x Conditional independence: for all x 1 ,x 2 ,x 3 , f X 1 ,X 3 | X 2 ( x 1 ,x 3 | x 2 ) = f X 1 | X 2 ( x 1 | x 2 ) f X 3 | X 2 ( x 3 | x 2 ) Review : HW6 Extra Problem #1 Uncorrelation: Cov( X,Y ) = E ( ( X E( X ))( Y E( Y )) ) = 0 independence uncorrelation, but not conversely
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lect10 - EE 278 Lecture Notes 10 Course Summary Basic...

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