When Y has positivelyskeweddistributionoverthepositiverealline,statisticalanalysesoften treat X = log(Y ) as havinga N(, 2) distribution. Then Y is

When Y has positivelyskeweddistributionoverthepositiverealline,statisticalanalysesoften treat X = log(Y ) as havinga N(μ, σ2) distribution. Then Y is saidtohavethe log-normal distribution.

(a) Deriveanexpressionforthe cdf G of Y in termsofthe cdf F of X, andtakethederivative to obtainthe pdf g of Y .

(b) UsetheinformationgiveninExercise2.66aboutthe mgf of anormalrandomvariable to showthat E(Y ) = eμ+σ2~2 and var(Y ) = [eσ2 − 1][E(Y )]2. Asshownforthegamma distribution inExercise2.45,thelog-normalhasstandarddeviationproportionaltothe mean.

(c) Explainwhythemedianofthedistributionof Y is eμ. Whatdothemeanandmedian suggest abouttheskewnessofthedistribution?

(d) Forindependentobservations y1, ...,yn from thelog-normal,wecouldsummarizethe distribution byfinding x for {xi = log(yi)} and thenusing exp(x). Showthat exp(x) =

(Πi yi)1~n, the geometricmean of {yi}.