Let (X,Y ) be randomvariablestakingvaluesin N0 R+ with a distributiondeterminedasfollows:X is drawnfromaPoissondistributionwithmean yielding thevalue x

Let (X,Y ) be randomvariablestakingvaluesin N0 × R+ with a distributiondeterminedasfollows:X is drawnfromaPoissondistributionwithmean

λ yielding thevalue x and subsequently Y is drawnfromagammadistributionwith shape parameter x + 1 and scaleparameter β. Inotherwords,thejointdistribution of (X,Y ) has density

with respectto m × ν, where m is countingmeasureon N0 and ν is thestandard Lebesgue measureon R+.

a) Arguethatthefamilyofdistributionswithunknown (λ, β) ∈ R2 +, mayberepres-
ented asaminimalandregulartwo-dimensionalexponentialfamily,determine the canonicalparameters,thecanonicalparameterspace,associatedcanonical statistics, andcumulantfunction.

b) Findthemeanandcovariancematrixfor (X,Y )⊤.

c) Showthatthesubfamilygivenbytherestriction λ = β is aminimalandreg-
ular one-dimensionalfamilyanddeterminethecanonicalparameters,associated canonical statistics,andcumulantfunction.

Transcribed Image Text:

f(x,s)(x,y) = = x! 3x+1x! e-/8, y> 0, x = 0,1,...