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distribution of Vn * ( ken ) ?( b ) The calf of Q - A isFisher information .is the limiting distribution of Z'n ?of *` - A is independent of O . )|~ = 5 . ( In other words , find the value of A . )and define* ( 1 ) < * ( 2 ) < ... < * ( n) to be the order statistics .where * ( 1 ) and * ( n) are the sample minimum and maximum , respectively .|PO ( 0 - 0 < 20 ) = \ ( 1 + 20 ) 7 / 2|2 . [ 10 marks ] Suppose that * 1 . ... . In are independent random variables with densityNote : In parts ( a ) and ( b ) , make sure to specify both the distribution and its parameters .( b ) Define En = ~ [exp ( 2*< ( ken + 1 ) ) - exp ( 2 *< ( ken ) ] where {kan] are defined as in part ( 2 ) . What\f ( ac ) = 1 - 1 20/ for - 1 < < < ]( a ) Define (kin ] to be integers such that Vn ( kin - 1 / 2 ) - 0 ( as ~ - 00 ) . What is the limiting13 . [ 10 marks ] Suppose that * 1 . .... In are independent random variables with common palfUse this information to find a 95% confidence interval for A of the form [ @ - A , @ + A ] when( a ) Show that the distribution of ! _ A is independent of O . ( Hint : Show that the distributionfor some unknown real -valued parameter & . A " good " estimator of @ is @ = (* ( 1 ) + * ( no ) ) / 2( b ) Give an estimator of the standard error of the MILE from part ( a ) based on the observed\f ( 20 ; O ) = = for 8 - 1 < < < Q + 17 1 - ( 1 - 20 ) 2 / 2 for ac > O\for ac < O