The Paretodistribution, introducedbytheItalianeconomistWilfredoParetoin1909to describe(onappropriatescales)incomeandwealth,isahighlypositively-skeweddistribution that has pdf f(y;) = ~y+1 for y 1 and aparameter

The Paretodistribution, introducedbytheItalianeconomistWilfredoParetoin1909to describe(onappropriatescales)incomeandwealth,isahighlypositively-skeweddistribution that has pdf f(y;α) = α~yα+1 for y ≥ 1 and aparameter α > 0.

(a) Showthat f is alegitimate pdf (i.e., nonnegativeandintegratingto1).

(b) Find E(Y ) for α > 1.

(c) The power-lawdistribution is acorrespondingdiscretedistributionthathas pmf f(y;α) = c~yα+1 for y = 1, 2, 3, ..., where c is theconstantforwhichtheprobabilities sum to1.Explainwhynoconstant c exists suchthat f(y) = c~y is a pmf or pdf. (Cases with α ≈ 1 describemanyphenomena,includingmusicalmeasuresofsuccesssuchasthe numberofliveperformancesofmusicalcompositionsthathavehadatleastoneperfor-

mance andthenumberofplaysperweekonastreamingserviceofsongsplayedatleast once.29)