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(10 points) Let ynumber of patients to seek medical care at a particular urgent care facility...

(10 points) Let ynumber of patients to seek medical care at a particular urgent care facility in a one day period, and assume

(10 points) Let ynumber of patients to seek medical care at a particular urgent care facility in a one day period, and assume that, conditionally on θ, y has a Poisson(θ) distribution, so that E(y|θ) var(y(9) t θ ~ Gamma(α, β), where α > 0 and β > 0 are both fixed and known. -0. Further suppose tha Find the prior predictive mean and variance for the number of patients today, E(y) and var(y)
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Answer #1

We have to find E(Y) and Var(Y)

We have given

E(Y|\theta)=\theta ,var(Y|\theta)=\theta

As theta is gamma(alpha,beta)

E(\theta)=\alpha*\beta,var(\theta)=\alpha*\beta ^2

E(Y)=E(E(Y|\theta))=E(\theta)=\alpha*\beta

Var(Y)-E(var(Y|θ))+1,ar(E(Y|θ))-E(0) + var@) = a *5+α * B2ー a* UaT UaT

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