# Concept derivation guides

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Concept guide



## [IID sequence: proving IID from the Bernoulli(1/3) cylinder product formula](/en/derivations/derivation-mko-iid-sequence-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-iid-sequence)

Let p=1/3, and let the cylinder certificate give each finite binary word w (length n, with k ones) the probability p^k(1−p)^{n−k}. Prove that this family is consistent with total mass 1 at every level; obtain from it (with the cited Kolmogorov existence theorem) a probability on {0,1}^ℕ; show that every X_i has the complete marginal Bernoulli(1/3) and that every finite subfamily is independent. Then use three contrasts to show that a checked prefix, pairwise independence and equal expectations are each insufficient for an infinite IID claim.

Independent identically distributed sequence

derivation-mko-iid-sequence-01





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