Pstat 127 HW3 Q4,5 sol.pdf - 4 Suppose Y1 Yn are independently distributed random variables with P(Yi = yi =(1 pi)yi pi if yi cfw_0 1 2 3 where pi(0 1(1

Pstat 127 HW3 Q4,5 sol.pdf - 4 Suppose Y1 Yn are...

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4. SupposeY1, . . . , Ynare independently distributed random variables withP(Yi=yi)=(1-pi)yipi,ifyi∈ {0,1,2,3, . . .}wherepi(0,1).(1)Suppose that we modelln(μi) =α+β xiwhereμi=E[Yi|xi]wherexiis the fixed known value of an explanatory variable “x” for theithindividual, fori∈ {1, . . . , n}.Treat the values of the explanatory variable as fixed and known for thenindividuals. Assume thatxi6=xjfor alli6=j.(a) Are the random variablesY1, . . . , Ynidentically distributed? (Answer Yes or No.) No. (b) Findμi(i.e.,E(Yi|xi)) as a function ofpi, based on the pmf above.Hint: one approach is to first find the moment generating function forYiand then use the MGF-based approach you studied inPSTAT 120A/B to find the mean. Otherwise you can use PSTAT 120A definition of mean for a discrete random variable directly.