4. (20 points) Consider a family of functions f[$) = karma at)\" where m > 0, n > 0 and is: is a constant chosen such that 1 f f (at) da: 2 1. 0 These functions represent a class of probability distributions, called beta distributions, where the proba- bility of a quantity a: lying between a and b (where 0 S a S b S 1) is given by PM, = f f($)dm. The median of a probability distribution is the value b such that the probability that b S :1: S 1 is equal m%=m% The expected value of one of these distributions is given by 1 f :1: f (2:) dx. I] Suppose information retention follows a beta distribution with m = 1 and n = % Consider an experiment where 2: measures the percentage of information students retain from their Calculus I course. 1. Find 19. 2. Calculate the probability a randomly selected student retains at least 50% of the information from their Calculus I course. 3. Calculate the median amount of information retained. 4. Find the expected percentage of information students retain