Exercise3.2 1. Let Y be acontinuousrandomvariablewithacdf F(y). Definearandomvariable Z = F(Y).

Showthat Z ∼ U (0,1) for any F(·).

2. Let Y1, ...,Yn be arandomsamplefromadistributionwithacdf Fθ (y), where θ is anunknown parameter.Showthat Ψ(Y;θ) = −Σni

=1 lnFθ (Yi) ∼ 1 2 χ2 2n and, therefore,isapivot.

(Hint: usingtheresultsofthepreviousparagraph,findfirstthedistributionof −lnFθ (Yi), relate it tothe χ2-distribution,andusethepropertiesofthelatter.)