Poisson random variable with a random mean: MMSE estimator, mode, mean, and MAP estimator Consider x = (x)no modeled as conditionally independent, identically, distributed Poisson random vari- ables given the mean parameter 1: 1 [n] = (v/[u]ac ) vlad "1{0,1,...} (2) for > > 0. We choose the exponential prior pdf for 1: fA(A) = Expon(X/b) = be-10, too) (1) where b > 0 is a known constant. (a) Find the posterior pdf fAIX ()|ac) given all measurements . ( b ) Compute the mean and mode of the posterior pdf, which are the Bayesian minimum mean-square error (MMSE) and maximum a posterior (MAP) estimates of 1, respectively.(c) Verify the obtained MAP expression by computing AMAP = arg max log px (x )) +log fA( X)]. Note that "log" denotes the natural logarithm