Random variables X1, . . . , Xs have the following joint probability density function: as the following picture.Question 1. (Graded, 15 points) Random variables X1, . . . , X5 have the following joint proba-bility density function: 1+:c1x2mccs when %5xi5 %,é=1,2…,s fX1,…,X3 (1171, . . . 51123) ={ 0 otherwise Answer the following questions and show su?icient steps to support your answers:a) Prove that fX1,…,Xs($’1s . . . ,ms) is indeed a valid PDF.b) Prove that the marginal distribution of Xi, for any 2′ = 1, 2, . . . , s, is Uniform(1/2, 1/2).0) Suppose s 2 4, are X1, X2,X3 independent?d) Are X1,X2,…,XS independent? e) Find the conditional PDF: f(X2:___,XS){X1=$1}(iE2, . . . ,malml) =?, given X1 = 1:1 E (1/2,1/2).
Random variables X1
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