Reparameterizingthegamma distribution byreplacing by k~, showthat f(y; k,) = (k~)k (k) eky~yk1, y 0 (2.17)

Reparameterizingthegamma distribution byreplacing λ by k~μ, showthat f(y; k,μ) =

(k~μ)k

Γ(k) e−ky~μyk−1, y ≥ 0 (2.17)

For this parameterization,showthattheexpressionsin(2.11)simplifyto E(Y ) = μ and σ =

μ~

º

k. So,forfixed k, asthe mean grows,sodoesthestandarddeviation.Thisisthecasein practice formanypositively-valuedvariables,suchasincome,inwhichgroupswithlargermean also tendtohavelargerstandarddeviation.