Sections5.1-5.2-students_SP11

Sections5.1-5.2-students_SP11 - Joint Distributions -Two or...

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1 Joint Distributions -Two or More RVs So far we discussed only a single RV In real (useful) experiments, we usually collect information on two or more quantities simultaneously Pressure, Volume, Temperature Service Times, # of A/C units # of cars and # of trucks on a Hwy fuel consumption and average speed traveled height and weight time until failure of two different components We need to be familiar with probabilistic modeling of two or more RV’s Prediction of one variable given the values of the other(s) Section 5.1 Joint Distribution for Discrete RV’s Def: The joint pmf of X and Y is a function p ( x , y ) or p XY ( x , y ) satisfying: Genuine joint pmf: Let A be a set of (x,y) pairs. Find the P(A): e.g. A = {(x,y): x + y = 5} or A = {(x,y): x ≥ 3, y ≤ 4}
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2 Example 5.1 X = auto policy deductible Y = homeowner’s policy deductible ...for a randomly selected person in an insurance pool Joint Probability Table Marginal Distributions The marginal pmf’s of X and Y, denoted by p X (x) and p Y (y) are
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3 Example -Transportation • The joint probability distribution (pmf) of the number of cars, X, and the number of buses, Y, per signal cycle at a proposed left turn lane (as a two way table) # buses y # cars p(x,y) 0 1 2 Col. Sum 0 .025 .015 .01 .05 1 .050 .030 .02 .10 x 2 .125 .075 .05 .25 3 .150 .090 .06 .30 4 .100 .060 .04 .20 5 .050 .030 .02 .10 Row Sum .5 .3 .2 1.00 a. Probabilities of some events of interest : P (exactly one car and one bus during a signal cycle) = p (1,1) = P (at most two cars at most one bus) = ( 2, 1) (2,1) P X Y F P(
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Sections5.1-5.2-students_SP11 - Joint Distributions -Two or...

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