Outline Outline 2 Probability Models of N Random Variables - PowerPoint PPT Presentation

Outline Outline 2 � Probability Models of N Random Variables � Probability Models of N Random Variables (5.1, Y&G) � Vector Notation (5.2, Y&G) � Vector Notation (5 2 Y&G) 204312 PROBABILITY AND 204312 PROBABILITY AND RANDOM PROCESSES FOR � Marginal Probability Functions (5.3, Y&G) COMPUTER ENGINEERS COMPUTER ENGINEERS � Independence of Random Variables and Random Vectors (5.4, Y&G) ( , ) Lecture 8: � Expected Value Vector and Correlation Chapters 5.1-5.4, 5.6 p Matrix (5.6, Y&G) Matrix (5.6, Y&G) 1st Semester, 2007 Monchai Sopitkamon, Ph.D. Probability Models of N Random Probability Models of N Random Variables I (5.1) Variables II 3 4 � Represent n RVs using vector notation � Multivariate joint PDF of the continuous RVs X 1 , …, X n is: � A random vector treats a collection of n RVs as a single ∂ ∂ … n F F ( ( x x , , x x ) ) … entity = X , , X 1 n f ( x , ..., x ) 1 n ∂ ∂ X , ..., X 1 n � 1 n x x � Perform an experiment that produces n RVs, X 1 , …, X n p p , 1 , , 1 n n defined with multivariate joint CDF � If X 1 , …, X n are discrete RVs with joint PMF = ≤ ≤ F ( ( x , ..., x ) ) P ( ( X x , ..., X x ) ) … X X , , X X 1 1 n n 1 1 1 1 n n n n P P P P ( ( 1 x , ..., x ) ) 1 1 n X , ..., X n � However, joint PMF/PDF provides a better way to 1 n ≥ 0 P P ( 1 x , ..., x ) 1. analyze prob models analyze prob. models X , ..., X n ∑ ∑ 1 1 n n = � P 1 . 2. � Multivariate joint PMF of the discrete RVs X 1 , …, X n is: … X , , X 1 n ∈ ∈ x S x S 1 X n X 1 n = = = … P ( x , ..., x ) P ( X x , , X x ) … X , , X 1 n 1 1 n n 1 n

Probability Models of N Random Vector Notation I (5.2) Vector Notation I (5.2) Variables III 5 6 � Use boldface notation x for a column vector. U b ldf i f l � If X 1 , …, X n are continuous RVs with joint PDF � Row vectors are transposed column vectors; x ′ . f f ( 1 ( 1 x , ..., x ) ) � Components of a column vector are written in a column. C t f l t itt i l X X , , ..., ..., X X n n 1 1 n ≥ f ( 1 x , ..., x . ) 0 1. � To save space, use the transpose of a row vector to X , ..., X n 1 n y ] ′ is a column represent a column vector: y represent a column vector: y = [ y 1 [ y 1 y n ] is a column � � ∫ ∫ x ∫ ∫ x = . j j � n … � F F ( ( x , ..., x ) ) f f ( ( u , , u ) ) d du d du 2. vector … … X , , X 1 n X , , X 1 n 1 n − ∞ − ∞ 1 n 1 n ∞ ∞ � A random vector is a column vector X = [ X 1 , …, X n ] ′ ∫ ∫ ∫ ∫ = 1 = 1 � � … � � 1 n f f ( ( x x , , x x ) ) dx dx dx dx . 3. where each X i is a RV … X , , X 1 n 1 n − ∞ − ∞ 1 n � Prob. of an event A expressed in terms of the RVs � A RV is a random vector with n = 1. X 1 , …, X n is � A sample value of a random vector is a column vector x = [ x 1 , …, x n ] ′ where the i th component, x i , of the ∑ = … P ( A ) P ( x , , x ) � Discrete: … X , , X 1 n vector x is a sample value of a RV X i vector x is a sample value of a RV X i 1 n ∈ A ( 1 … ( x , , x n ) ) A � Continuous: ∫ ∫ = � … … P ( A ) f ( x , , x ) dx dx dx … X , , X 1 n 1 2 n 1 n A Vector Notation II Vector Notation II Vector Notation III Vector Notation III 7 8 � Random Vector Probability Functions � Probability Functions of a Pair of Random Vectors: For random vectors X with n components and Y with � CDF of a random vector X is m components = F ( x ) F ( x ,..., x ) X X ,..., X 1 n � The joint CDF of X and Y is 1 n = � PMF of a discrete random vector X is F ( x , y ) F ( x ,..., x , y ,..., y ) X , Y X ,..., X , Y ,..., Y 1 n 1 m 1 n 1 m = � The joint PMF of discrete random vectors X and Y is P ( x ) P ( x ,..., x ) X X ,..., X 1 n 1 n = P ( x , y ) P ( x ,..., x , y ,..., y ) X X , , Y Y X X ,..., ,..., X X , , Y Y ,..., ,..., Y Y 1 1 n n 1 1 m m � PDF of a continuous random vector X is PDF f ti d t X i 1 1 n n 1 1 m m � The joint PDF of continuous random vectors X and Y is = f f ( ( x ) ) f f ( ( x ,..., , , x ) ) X X X X ,..., X X 1 1 n n = 1 n f ( x , y ) f ( x ,..., x , y ,..., y ) X , Y X ,..., X , Y ,..., Y 1 n 1 m 1 n 1 m

Vector Notation IV Vector Notation IV Vector Notation V Vector Notation V 9 10 � Ex 5 4 Random vector X has PDF � Ex.5.4: Random vector X has PDF ∫ ∫ ∫ ∫ ∫ ∫ x x x x x x − − − = 3 3 2 2 1 1 x 2 x 3 x F ( x ) 6 e dx dx dx 1 2 3 ⎧ ′ − ≥ X 1 2 3 a x 6 e x 0 0 0 0 = ⎨ f ( x ) X � ⎩ ⎩ 0 0 otherwise th i where a = [1 2 3] ′ . What is the CDF of X ? � � Since a has 3 components, X should be a 3- � dimensional random vector. Expanding a ′ x , write � � the PDF as a function of the vector components, � ⎧ − − − ≥ x 2 x 3 x 6 e x 0 1 2 3 = = ⎨ ⎨ i f f X x ( ( x ) ) ⎧ − − − − − − ≥ x 2 x 3 x ⎩ 0 otherwise ( 1 e )( 1 e )( 1 e ) x 0 1 2 3 = ⎨ i F ( x ) Find CDF of X by integrating the PDF w/ respect to / X ⎩ ⎩ 0 0 otherwise otherwise the 3 variables to obtain Marginal Probability Functions II Marginal Probability Functions II Marginal Probability Functions I (5.3) Marginal Probability Functions I (5.3) 11 12 � Study some of the RVs and ignore other ones. � For a joint PDF f W, X, Y, Z ( w , x , y , z ) of continuous RVs W, X, Y, Z , some marginal PDFs are � For a joint PMF P W X Y Z ( w , x , y , z ) of discrete RVs j W, X, Y, Z ( , , y , ) W, X, Y, Z , some marginal PMFs are ∑ ∑ = = ∞ P P ( ( x x , y y , z z ) ) P P ( ( w w , x x , y y , z z ) ) ∫ ∞ ∫ = f f ( ( x , y , z ) ) f f ( ( w , x , y , z ) ) d dw X , Y , Z W , X , Y , Z X , Y , Z W , X , Y , Z ∈ w S - W ∑ ∑ ∑ ∑ = = ∞ ∞ P P ( ( w w , z z ) ) P P ( ( w w , x x , y y , z z ) ) ∫ ∫ ∫ ∫ = f f ( ( w , z ) ) f f ( ( w , x , y , z ) ) d dx dy d W , Z W , X , Y , Z W , Z W , X , Y , Z ∈ ∈ − ∞ − ∞ x S y S X Y ∑ ∑ ∑ ∑ ∑ ∑ = ∞ ∞ ∞ P P ( ( x x ) ) P P ( ( w w , x x , y y , z z ) ) ∫ ∫ ∫ ∫ ∫ ∫ = f f ( ( x ) ) f f ( ( w , x , y , z ) ) d dw dy d dz d X X W W , X X , Y Y , Z Z X W , X , Y , Z ∈ ∈ ∈ − ∞ − ∞ − ∞ w S y S z S W Y Z