18.44 Lecture Notes - Lecture 15: Random Variable
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Let x be such that the rst heads is on the xth toss. For example, if the coin sequence is tththt, then x = 3. , where q =1 p is tails probability. Say x is a geometric random variable with parameter p. Let x be a geometric with parameter p, i. e. , p{x = k} = There"s a trick to computing sums like this. Let x be a geometric random variable with parameter p. then p{x = k} = Let"s try to come up with a similar trick. Thus e[(x 1) ] = e[x 2x + 1] = e[x ] 2e[x] + 1= e[x ] 2/p + 1 = qe[x ]. Solving for e[x ] gives (1 q)e[x ] = pe[x ] = 2/p 1, so e[x ] = (2 p)/p . Setting j = k 1, we have e[(x 1) ] = Var[x] = (2 p)/p 1/p = (1 p)/p = 1/p 1/p = q/p .
