6. Let X1, X2, X3 be a random sample from a distribution. Hint: Derive the joint pdf of Y₁, Y2, Y3 in parts (a) and (b). Also, note the following relation between th joint pdf of order statistics X (1), X (2),..., X (n) and the joint pdf of the corresponding random sample X1, X2, Xn f(x(1), x(2),..., x(n))=n! f(x1, x2,...,xn). - a. X Exponential (B). Prove or disprove that Y₁ = X (1) X(2) , Y₂ = X(2) X(3) , and Y3 = X(3) are independent, where X(), i=1,2,3 denotes the ith order statistic. - b. X f(x)= axa-1 0< x < 0, a >0. ' θα Prove or disprove that Y₁ = X(), Y2 the ith order statistic. X(2) X(2) X(3) = , and Y3 = X(3) are independent, where X), i=1,2,3 denote: X (1) X(2) ,Y₂ = X(2) X(3) , Y3 = X (3) X(4) , Yn-1 = X (n-1), and X (n) are independent? c. Based on your conclusions for parts (a) and (b), would you claim that if the pdf of X is a member of scale parameter pdfs, then Y₁ = Explain. X(n)