You are part of a research team that focuses on a rare species of butterfly, whose pre-rain flight patterns have been observed to correlate with rainfall amounts. The rate at which the butterfly flaps its wings, flutterness, affects the rainfall amounts. You have developed a statistical model, called the Flutter Distribution, to encapsulate this phenomenon. This model leverages parameter, , to quantify the observed flutterness, x. The Flutter Distribution is delineated by the probability density function (PDF) where x [0, 1] and > 0: p(x|) = (1 x)^1

1.a Based on a dataset X = {x1, x2, . . . , xn} of flutterness measurements, determine is the Maximum Likelihood Estimate (MLE) the parameter .

1.b Your team determined the prior of as p() = ^(1)e^() where > 0. Find the Maximum a Posteriori (MAP) estimation of the parameter based on p().