1. practice with priors bookmark this page one side concept introduced in the second bayesian lecture is the conjugate prior. simply put, a prior distribution π(θ) is called conjugate to the data model, given by the likelihood function l(xi|θ) , if the posterior distribution π(θ|x1,x2,…,xn) is part of the same distribution family as the prior. this problem will give you some more practice on computing posterior distributions, where we make use of the proportionality notation. it would be to try to think of computations in forms that are reduced as much as possible, as this will with intuition towards assessing whether a prior is conjugate. this problem makes use the gamma distribution (written as gamma(k,θ) ) is a probability distribution with parameters k> 0 and θ> 0 , has support on (0,[infinity]) , and whose density is given by f(x)=xk−1e−xθγ(k)θk . here, γ(k)=∫[infinity]0tk−1e−tdt is the euler gamma function . (a) 3 points possible (graded) suppose we have the prior π(λ)∼ exp(a) (where a> 0 , and conditional on λ , we have observations x1 , x2 , ⋯ , xn ∼i. i.d exp(λ) . compute the posterior distribution π(λ|x1,x2,…,xn) . the posterior distribution for λ is a gamma distribution. what are its parameters? enter your answer in terms of a , n , and ∑i=1nxi (enter sigma_i(x_i) for ∑i=1nxi . do not worry if the parser does not render properly; the grader works independently. if you wish to have proper rendering, enclose sigma_i(x_i) by brackets. )