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# ch1pt3 - Chapter 1 Review of Probability Random Variable...

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Chapter 1: Review of Probability & Random Variable Concepts Multiple Random Variables Dr. Lim HS Last Updated: 2 June 2009 c 2009 MMU

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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

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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 unit-step 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. 0 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 } ≥ 0 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

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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

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Joint CDF and PDF Joint CDF (cont.) The marginal cdf’s are plotted in Figure 2. 1 2 3 x F X ( x ) 0.2 0.5 1.0 1 2 3 y F Y ( y ) 0.5 1.0 Figure 2: Marginal cdf’s. c 2009 MMU Page 7/60
Joint CDF and PDF Joint PDF For two rv’s X and Y , the joint probability density function (pdf) is defined as f X,Y ( x, y ) = 2 F X,Y ( x, y ) ∂x∂y For two discrete rv’s X and Y , we have f X,Y ( x, y ) = X n X m P ( x n , y m ) δ ( x - x n ) δ ( y - y m ) In general, f X 1 ,X 2 ,...,X N ( x 1 , x 2 , ..., x N ) = N F X 1 ,X 2 ,...,X N ( x 1 , x 2 , ..., x N ) ∂x 1 ∂x 2 ...∂x N c 2009 MMU Page 8/60

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Joint CDF and PDF Joint PDF (cont.) By direct integration, we may find the distribution function F X 1 ,X 2 ,...,X N ( x 1 , x 2 , ..., x N ) = Z x N -∞ · · · Z x 2 -∞ Z x 1 -∞ f X 1 ,X 2 ,...,X N ( ξ 1 , ξ 2 , ..., ξ N ) 1 2 ...dξ N c 2009 MMU Page 9/60
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ch1pt3 - Chapter 1 Review of Probability Random Variable...

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