ioe265f11-Lec13 - Lec 13 Joint Probability Distributions...

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Lec 13 - Joint Probability Distributions IOE 265 F11 1 1 Jointly Distributed Random Variables 2 Topics I. Jointly Distributed Variables II. Joint Distributions of Unrelated Variables Two Independent Random Variables III. Expected Values of Joint Distributions IV. Joint Distributions of Related Variables Covariance and correlation measures of “degree of association”
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Lec 13 - Joint Probability Distributions IOE 265 F11 2 3 I. Jointly Distributed Variables Many problems in statistics and probability involve more than a single random variable. Therefore, sometimes it is necessary to study several random variables simultaneously. X Y Z 1 0 0 0 1 1 0 1 0 1 0 1 X = height (1 in spec, 0 out of spec) Y = width (1 in spec, 0 out of spec) Z = depth (1 in spec, 0 out of spec) X Y Z 4 Types of Jointly Distributed Variables Two Discrete Joint pmf Marginal pmf (consider 1 of 2 joint variables) Conditional pmf Two Continuous Joint pdf Marginal pdf (consider 1 of 2 joint variables) Conditional pdf Independent variables More than two variables
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Lec 13 - Joint Probability Distributions IOE 265 F11 3 5 Joint Distribution – Two Discrete Joint pmf Let X and Y represent 2 discrete rv’s on space S p XY (x,y) >= 0 p XY (x,y) = P(X=x, Y=y) Marginal pmf To obtain a marginal pmf for say X=100, P(100,y) - you compute prob for all possible y values. x y y x y x p y p y x p x p ) , ( ) ( ) , ( ) ( 1 ) , (  x y XY y x p 6 5-1.3 Conditional Probability Distributions 5-1 Two Discrete Random Variables
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Lec 13 - Joint Probability Distributions IOE 265 F11 4 7 Example - Joint Probability TV Brand Example (repeated from conditional probability lecture notes) Event A: sell a TV from one of 3 brands (A1, A2 or A3); Event B: repair TV Suppose Selling Mix: A 1 = 50%, A 2 = 30% and A 3 = 20% Likelihood to Repair Given Model A 1 = 25% Likelihood to Repair Given Model A 2 = 20% Likelihood to Repair Given Model A 3 = 10% First Convert information to joint probability table. Example: p(x=A 1 , y=repair) = 0.5*0.25 = 0.125 8 Joint Probability Table What are some requirements of joint probability table?
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  • Fall '07
  • Jin
  • Probability theory, Joint probability distribution, joint probability distributions, Mixture Experiments, Joint Dist.

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