Consider flipping a strange coin multiple times. For this strange coin, (a) Pr[heads in toss k] = p if k is odd (b) Pr[heads in toss k] = q if k is even Let X be the number of times this strange coin must be tossed until we get the first heads. Recall that in the special case that q = p, X Geo(p). Let us calculate the PDF for X in the general case when q = p. In particular, prove that, Pr[X = k] = (1 p) k1 2 (1 q) k1 2 p if k is odd Pr[X = k] = (1 p) k 2 (1 q) k 2 1 q if k is even [15 marks] Now that we have the PDF for this strange coin, we can calculate its expected value. If r := (1 p) (1 q) and R Geo(1 r), prove that, E[X] = 2p 1 r E[R] p 1 r + 2q (1 p) 1 r E[R] [25 marks] Using the above result, prove that in the special case when q = p, E[X] =