ECE313.Lecture32

ECE313.Lecture32 - ECE 313 Probability with Engineering...

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Jointly Continuous Random Variables I Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 Probability mass on the real line The pmf p X (u) for a discrete random variable X describes a collection of point masses on the real line: P{ X = u} = p X (u) A continuous random variable X spreads the mass on (an interval of) the real line There is no probability mass at any point P{ X = u} = 0 for all real numbers u pdf f X (u) is the density of the probability mass Units are probability mass per unit length P{ X { small interval that contains number u}} ≈ f X (u)•{length of the interval}
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 Joint behavior of continuous RVs When X and Y are discrete RVs, the random point ( X , Y ) is also discrete valued The joint pmf p X , Y (u,v) describes point masses in the plane If X and Y are continuous RVs, then either ( X , Y ) can take on all possible values in a region of nonzero area or Y = g( X ), and thus ( X , Y ) always lies on the curve v = g(u) in the plane
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 Probability mass in the plane Suppose that X and Y are continuous RVs, and the random point ( X , Y ) can take on all values in a region of nonzero area In this case, X and Y are said to be jointly continuous RVs The probability mass is spread over this region of the plane If Y = g( X ), then the probability mass is spread along the curve v = g(u) X and Y are not jointly continuous RVs
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 Probability mass along a curve If Y = g( X ), then the probability mass is spread along the curve v = g(u) X and Y are not jointly continuous RVs All questions involving the probabilistic behavior of the random point ( X , Y ) can be translated into questions involving X alone Example: If Y = X 2 , then for any v ≥ 0, the joint CDF of X and Y is: F X , Y (u,v) = P{ X ≤ u, Y ≤ v} = P{ X ≤ u, X 2 ≤ v} = P{–a ≤ X ≤ a} where a = min{u, v}
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 Probability mass over a region The case of X and Y being continuous random variables but not jointly continuous random variables is easily treated via the methods that we have studied previously The interesting case is when X and Y are jointly continuous and the probability mass is spread over a region of the plane Abuse of language: We will assume that, unless explicitly stated otherwise, continuous RVs are also jointly continuous
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ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved
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ECE313.Lecture32 - ECE 313 Probability with Engineering...

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