20 points Let X be a nonnegative integer valued random variable (i.e., X = {0, 1,...

 

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20 points Let X be a nonnegative integer valued random variable (i.e., X = {0, 1, 2, . . .}). We will prove that: E[X] = P[X > k] = P[X > k]. k=1 k=0 (1) (a) First, consider the random variables I0, 11, 12,... where Ik = Ik(X). Ik(x) is the function: (1, if x > k. Ik(x) = if x < k. (b) Prove that E[Ik] = P[X > k]. (c) Write X in terms of an infinite sum of the random variables I, 12, ... (d) Prove that E[X] = P[X > k]. k=0 20 points Let X be a nonnegative integer valued random variable (i.e., X = {0, 1, 2, . . .}). We will prove that: E[X] = P[X > k] = P[X > k]. k=1 k=0 (1) (a) First, consider the random variables I0, 11, 12,... where Ik = Ik(X). Ik(x) is the function: (1, if x > k. Ik(x) = if x < k. (b) Prove that E[Ik] = P[X > k]. (c) Write X in terms of an infinite sum of the random variables I, 12, ... (d) Prove that E[X] = P[X > k]. k=0