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Unformatted text preview: Chapter 1: Review of Probability & Random Variable Concepts Multiple Random Variables Dr. Lim HS Last Updated: 2 June 2009 c 2009 MMU Presentation outline ◦ Joint CDF and PDF ◦ Conditional Distributions ◦ Statistical Independence ◦ Expectation and Moments ◦ Jointly Gaussian RV’s ◦ Transformation of Multiple RV’s c 2009 MMU Page 1/60 Joint CDF and PDF • Joint CDF ◦ The joint cumulative distribution function (cdf) of two rv’s X and Y is defined as F X,Y ( x,y ) = P { ( X ≤ x ) ∩ ( Y ≤ y ) } = P { X ≤ x,Y ≤ y } Figure 1: Comparison of events in S with those in S J . c 2009 MMU Page 2/60 Joint CDF and PDF • Joint CDF (cont.) ◦ A rv X may be thought of as a random point on the real line. Likewise, a pair of rv’s ( X,Y ) can be thought of as a random point in the x y plane. ◦ For two discrete rv’s X and Y F X,Y ( x,y ) = X n X m P ( x n ,y m ) u ( x x n ) u ( y y m ) where P ( x n ,y m ) is the probability of the joint event { X = x n ,Y = y m } and u ( . ) is the unitstep function. ◦ In general, for N rv’s X n , n = 1 , 2 ,...,N , F X 1 ,X 2 ,...,X N ( x 1 ,x 2 ,...,x N ) = P { X 1 ≤ x 1 ,X 2 ≤ x 2 ,...,X N ≤ x N } c 2009 MMU Page 3/60 Joint CDF and PDF • Joint CDF (cont.) ◦ Important properties of the joint cdf: 1. F X,Y (∞ ,∞ ) = 0 F X,Y (∞ ,y ) = 0 F X,Y ( x,∞ ) = 0 2. F X,Y ( ∞ , ∞ ) = 1 3. ≤ F X,Y ( x,y ) ≤ 1 4. F X,Y ( x,y ) is a nondecreasing function of both x and y 5. F X,Y ( x 2 ,y 2 ) + F X,Y ( x 1 ,y 1 ) F X,Y ( x 1 ,y 2 ) F X,Y ( x 2 ,y 1 ) = P { x 1 < X ≤ x 2 ,y 1 < Y ≤ y 2 } ≥ 6. F X,Y ( x, ∞ ) = F X ( x ) F X,Y ( ∞ ,y ) = F Y ( y ) ◦ F X ( x ) (or F Y ( y ) ) obtained in Property 6 are called the marginal cdf. c 2009 MMU Page 4/60 Joint CDF and PDF • Joint CDF (cont.) Example 1. Show that F X ( x ) can be obtained by setting the value of y to infinity in F X,Y ( x,y ) . Recall that F X,Y ( x,y ) = P { X ≤ x,Y ≤ y } = P ( A ∩ B ) where A = { X ≤ x } and B = { Y ≤ y } . Now if y = ∞ , B = { Y ≤ ∞} = S , thus F X,Y ( x, ∞ ) = P ( A ∩ S ) = P ( A ) = P { X ≤ x } = F X ( x ) c 2009 MMU Page 5/60 Joint CDF and PDF • Joint CDF (cont.) Example 2. Assume that the joint sample space S J has only three possible elements: (1,1), (2,1), and (3,3). The probabilities of these elements are to be P (1 , 1) = 0 . 2 , P (2 , 1) = 0 . 3 , and P (3 , 3) = 0 . 5 . The joint distribution function is F X,Y ( x,y ) =0 . 2 u ( x 1) u ( y 1) + 0 . 3 u ( x 2) u ( y 1) + 0 . 5 u ( x 3) u ( y 3) If we set y = ∞ , F X,Y ( x, ∞ ) = F X ( x ) = 0 . 2 u ( x 1) + 0 . 3 u ( x 2) + 0 . 5 u ( x 3) If we set x = ∞ , F X,Y ( ∞ ,y ) = F Y ( y ) = 0 . 2 u ( y 1) + 0 . 3 u ( y 1) + 0 . 5 u ( y 3) = 0 . 5 u ( y 1) + 0 . 5 u ( y 3) c 2009 MMU Page 6/60 Joint CDF and PDF • Joint CDF (cont.) The marginal cdf’s are plotted in Figure 2....
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This note was uploaded on 08/21/2009 for the course FET 44 taught by Professor ;im during the Spring '09 term at Multimedia University, Cyberjaya.
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