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Unformatted text preview: Jointly Continuous Random Variables II Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 2 of 35 Jointly continuous RVs Jointly continuous random variables spread the probability mass (usually with varying density ) over a region The joint pdf f X , Y (u,v) tells us how dense the probability mass is at the point (u,v) There is no probability mass at any point P{( X , Y ) = (u,v)} = P{ X = u, Y = v} = 0 for all real numbers u and v f X , Y (u,v) is the density of the probability mass Units are probability mass per unit area ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 3 of 35 The joint pdf is mass per unit area The joint pdf f X , Y (u,v) is not a probability: we must multiply the pdf by an area to get a probability P{( X , Y ) ∈ { small region containing (u,v)}} ≈ f X , Y (u,v)•{area of region} For larger regions, Probability = (double) integral of the pdf over the region P{( X , Y ) ∈ A} = ∫ ∫ f X , Y (u,v) du dv A ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 4 of 35 Graphical interpretation P{( X , Y ) ∈ { small region containing (u,v)}} ≈ f X , Y (u,v)•{area of region} ≈ volume above the uv plane and below the f X , Y surface u v Joint pdf is a surface above the uv plane height ≈ f X , Y (u,v) base area ≈ height × base area ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 5 of 35 Probability = volume under joint pdf P{( X , Y ) ∈ A} = ∫ ∫ f X , Y (u,v) du dv A P{u ≤ X ≤ u+ δ u,v ≤ Y ≤ v+ δ v} ≈ f X , Y (u,v)• δ u• δ v is the volume of a prism of height f X , Y (u,v) and rectangular base of area δ u• δ v P{( X , Y ) ∈ A} = volume of solid with vertical sides, base A, and varying height ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 6 of 35 Properties of the joint pdf f X , Y (u,v) ≥ 0 for all u and v Total volume between the f X , Y surface and the uv plane is 1 This is just the interpretation of the result F X , Y ( ∞ , ∞ ) = 1 = ∫ ∫ f X , Y (u,v) du dv v=– ∞ ∞ u=– ∞ ∞ F X , Y (u ,v ) = ∫ ∫ f X , Y (u,v) du dv or integrate w.r.t. v ¡rst and then w.r.t. to u v=– ∞ v u=– ∞ u ECE 313  Lecture 33 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 7 of 35 Finding joint pdf from the joint CDF f X , Y (u,v) = F X,Y (u,v) if the derivative exists, and f X , Y (u,v) = any number ≥ 0 otherwise ∂...
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 Spring '09
 mr.pil
 Probability theory, All rights reserved, Cumulative distribution function, Dilip V. Sarwate

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