Exercise2.17 An aircraftlaunchedamissiletowardsagroundtarget.Let X and Y be southnorth and eastwestrandomdeviationsofthemissilefromthetarget,respectively.Itmaybeassumedthat X and Y are i.i.d N (0,2).

Exercise2.17 An aircraftlaunchedamissiletowardsagroundtarget.Let X and Y be south–north and east–westrandomdeviationsofthemissilefromthetarget,respectively.Itmaybeassumedthat X and Y are i.i.d N (0,σ2). Define R =

√

X2+Y2 to bethedistancebetweenthetargetandthe actual locationofthemissile’sexplosion.

1. Showthatthedistributionof R has thedensity fσ (r) = r σ2 e− r2 2σ2 , r ≥ 0 (Rayleigh distribution).Doesitbelongtotheexponentialfamily?
(Hint: firstfindthedistributionof R2 = X2+Y2.)
2. Find theMLEof σ2.
3. Is theMLEunbiased?IsitUMVUE?