Question 1. (Graded, 15 points) Random variables X1, . , X5 have the following joint proba- bility density function:c1x2mccs when %5xi5 %,=1,2.,s...

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 suflicient 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).