Chapter5.10 lecture notes

# Chapter5.10 lecture notes - Chapter 5.10 JOINT...

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Chapter 5.10 JOINT DISTRIBUTIONS/ DISCRETE and Continuous I. (P 173 -175) Discrete Variables: (a) Joint distribution of X 1 and X 2 : f(x 1 ,x 2 ) = Pr(X 1 = x 1 , X 2 = x 2 ) (b)Marginal Distributions of X 1 and X 2 : f (x 1 ) = Pr ( X 1 =x 1 ) and f 2 (x 2 ) = Pr ( X 2 =x 2 ). Here f 1 (x 1 )= Σ f(x 1 , x 2 ) (sum over all values of x 2 ). (c) Conditional probability distribution of X 1 given X 2 : f(x 1 | x 2 ) = : f(x 1 ,x 2 )/ f 2 (x 2 ) for all x 1 ,x 2 provided f f 2 (x 2 ) 0. (d) Independent random variables : X 1 , and X 2 are independent if and only if f (x 1 ,x 2 ) = f 1 (x 1 ) f 2 (x 2 ). (e) Covariance of two random variable X 1 and X 2 = Cov(X 1 , X 2 ) = E(X 1 - μ 1 ) (X 2 - μ 2 ) . If X 1 and X 2 are independent then Cov(X 1 , X 2 ) =0. (f) Correlation between two random variables ρ 12 = Cov(X 1 , X 2 ) / σ 1 σ 2 where σ j denotes the standard deviation of X j for j=1,2. If X 1 , X 2 are independent then ρ 12 = 0.0. Range for ρ 12 is from -1 to +1. Here | ρ 12| =1.0 implies that X 2 = α + β X 1 and Var(X 2 |X 1 ) =0.0, Note also that ( Cov(X 1 , X 2 ) = ρ 12 σ 1 σ 2. ). Example: Two scanners are needed for an experiment . Of the five available 2 have electronic defects, another one has a defect in the memory and two are in good working order. Two units are selected at random. (a) find the joint

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