2 22 Let X1 X2 Xn denote the order statistics of a sample from a continuous unspecified distribution FX Define the n random variables Vi FX Xi FX Xi1 for 14i4n 1 and Vn FX Xn a Find the marginal distribution of Vr 14r4n b Find the joint distribution of Vr and FX Xr1 14r4n 1, and show that they are independent c Find the joint distribution of V1 V2 Vn d Show that V1 V2 Vn are independent e Show that V1 V2

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Question: =+2.22. Let X1 < X2 < < Xn denote the order statistics of a sample from a continuous unspecified distribution FX . Define the n

=+2.22. Let Xð1Þ < Xð2Þ < < XðnÞ denote the order statistics of a sample from a continuous unspecified distribution FX . Define the n random variables Vi ¼ FX ðXðiÞÞ

FX ðXðiþ1ÞÞ for 14i4n 1 and Vn ¼ FX ðXðnÞÞ

ðaÞ Find the marginal distribution of Vr; 14r4n.

ðbÞ Find the joint distribution of Vr and FX ðXðr1ÞÞ; 14r4n 1, and show that they are independent.

ðcÞ Find the joint distribution of V1; V2; ... ; Vn.

ðdÞ Show that V1; V2; ... ; Vn are independent.

ðeÞ Show that V1; V2

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