Suppose that x is the number of pregnant women arriving at a hospital at hayward, ca, to deliver their babies during a given month. the discrete count nature of the data and its natural interpretation as an arrival rate suggest adopting a poisson likelihood ,
f(xi0)=(e^-θ.θ^x)/x! , x=0, 1,2,…, θ> 0

to implement a bayesian analysis, a prior distribution is required for θ having support on the positive real line. a reasonable flexible choice is provided by the gamma distribution ,
fθ (θ)= (θ^α-1.e^θ/β)/ t(α)β^α , θ> 0, α > 0, β > 0,
or saying θ ~ garnma(α, β), where the values of α and β are known

(1) using bayes theorem to obtain the posterior density function of 0 (hint: use the proportional argument and identify the corresponding distributional kernel function
(2) (yes/no question) based on (1), is the gamma the conjugate family for the poisson likelihood?
(3) suppose we observe 42 moms arriving at our hospital to deliver babies during de- cember 2017. suppose we adopt the prior as gamma(5,6), which has mean 5(6)-30 and variance 5(6*6)-180, reflecting the hospital's totals for the past 12 months, which have been slightly less busy on average. use r, plot the posterior and prior density functions in one figure together
(4) based on (3), calculate the exact" 95% equal tail and highest posterior credible intervals for the parameter θ
(5) based on (3), use r programs "jags" and "rjags" to obtain the 95% equal tail and highest posterior density (hpd) credible intervals for the parameter θ using the simulated samples generated by gibbs sampler, a basic mcmc algorithm. (set up the model tuning parameters, e. g., burn in period, iterations by yourself and attach your r-code at the end of your homework.)