MATH 418 Midterm: MATH 418+544 2013 Winter Test 1
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Find functions f1, f2 : r r so that fi(x) are both uniform in [0, 1] and independent: repeat with an in nite sequence of functions fi so that fi(x) are all independent. For two random variables x, y , prove that. + 2 var(y )kxk2 (recall kxk is the minimal a so that p(|x| a) = 1). ) Xn which are symmetric and not constant (x and x have the same distribution), let sn = pi n xi. Partial marks will be given for proofs with some assumptions on the variables. Find the expected time until the sequence htthtthtth appears as follows: 2i, where i an if the rst i: consider the sequence mn = n pi an letters of the sequence are the results of coins n i + 1, . Mn is a martingale: use this to nd the required time. Justify all steps: write your answers for a general pattern if ypossible.
