Let X be a random variable representing the outcome of a biased coin with possible outcomes A = {0, 1}, x E X. The bias of the coin is itself controlled by a random variable O, with outcomes- 0 6 0, where 0 = (0 ER : 0 oo. 1For example, asserting that I; 2:2 ($3 + 23) air = 2/3 with no working out is adequate, as you could just plug the integral into Wolfram Alpha using the command Integrate [x'2Cx'3 + 2x) ,{x,0,1}] 2For example, a value of 9 = 1 represents a coin with 1 on both sides. A value of 6 = 0 represents a coin with O on both sides, and 9 = 1/2 represents a fair, unbaised coin. 3The coin ips are independent and identically distributed (i.i.d). 4'We write 331:,-l as shorthand for the sequence 22123:; . . . at\". e) Compute the variance 0?, of the distribution of {7' after observing n consecutive ones, with a uniform prior p(6) : 1. Provide intuition explaining the behaviour of 0,2, as n > 00. f) Compute the maximum or posteriom' estimation 6114,43, of the distribution on 6' after observing n consecutive ones, with a uniform prior 39(9) : 1. Provide intuition explaining how 6114A pn varies with n. g) Given we have observed an consecutive coin ips of ones in a row, what do you think would be a better choice for the best guess of the true value of 9? an or 6114A p? Justify your answer. (Assume pa?) = 1.) h) Plot the probability distributions P(9|$1:n : 1) over the interval 0 S 19 S 1 for n E {0, 1, 2, 3, 4} to compare them. Assume 39(9) : 1