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diff --git a/houdre/03_foundations b/houdre/03_foundations new file mode 100644 index 0000000..03d76da --- /dev/null +++ b/houdre/03_foundations @@ -0,0 +1,26 @@ +A probability space is Omega, an event space, and P: E -> [0,+\infty) + +Why do we use infinite sums (A_1...A_\infty) instead of just finite sum? + +Ex. relevant question: I have a fair coin and I flip it until I get a +tails. Are you sure you'll eventually get tails? + +Yes. + +Lemma: What are the odds that n coin flips result in no tails? +Complement: What are the odds that n coin flips result in only heads? + +Each flip is independent and fair, so: + +P(A_1 \cap A_2 \cap \ldots \cap A_n) = 1/2^n +P("""^C) = 1 - 1/2^n + +Using an infinite union A_1...A_\omega = lim_n->\infty P(A_1\cap...A_n) + +This axiom defines this relation. + +Properties of probabilities (can be derived from original axioms): + For all A \in E, P(A^C) = 1-P(A) + For all A,B \in E, if A \subset B, P(A) \leq P(B) + For all A,B \in E, P(A \cup B) = P(A)+P(B)-P(A\cap B) + Can construct Poincare lemma or inclusion/exclusion. |