Question

1. The chance of heads of a random coin is picked according to the beta (1.1) distribution. For each part, you must show work or reference a relevant section of the textbook. a) Find P(first toss is heads). b) Find P(first toss is heads, second toss is tails). c) Compare your answer in part b to the chance of {first toss is heads, second toss is tails given that you are tossing a fair coin twice. Is it larger or smaller? Give an intuitive explanation. d) Find the posterior density of the chance of heads given that you've seen one head and one tail.

Answer 1

chance of follow Heads of a randou coin Beta Chi) aksion. m ) Let x denote the chance of theaels. f = x (x ocael - ßemon) mzo nzo B(1, 1) - mal, nel = f(x) = but I cay (1) one as B (mina f (x ) (1.7 dx - flo) a fi dx 0<xel = flow is omform distby o cxcl a 60112 so P[ first chance is heads) = E(X) = a +b + = IP first chance is head) = x

head, secad chance pl first chance is a is tails As we Chow event's are Independent in toss of a coin = Pl fret chance Estheed] x Plans chances tail P ( secad Chance is tail] As we know xuß, (min) [x denote chance of stered] 1-xu Bilhim) {(1-x) denote Chance of tails] - 1 x uß (1,D as IME NED - 1-X u ulo } = E(1-x) = 2 [P{ chance g tail)] o ponto [first choice is head, The chance'is tacit]

when tossing a fair wen twice sample space - { HH, HT, TH, IT] P[ HT} = This is same as we obtain in case (b) Prob. density of chance of Heads quen that you have seen one head and one tail f [xlx, Ex] = f(x) As they are Independent events so posterior density of x B (11) distono is again