Math 6710 Lec 01

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<ul><li><p>7/30/2019 Math 6710 Lec 01</p><p> 1/2</p><p>Sean LiMath 6710 Notes Fall 2013</p><p>Grad Probability I</p><p>Lecture 1 8/28/13</p><p>Course Info</p><p>John Pike, jpike@cornell.edu</p><p>OH: MLT 580, WF: 1:30 - 2:30</p><p>Website: http://www.math.cornell.edu/~web6710/</p><p>Grading: All homework.</p><p>Probability is a measure space (, F, P) with P() = 1.</p><p> represents all possible outcomes of something. F is a -algebra, 2. It is nonempty,closed under complements, closed under countable unions. P is a set function F [0, 1] s.t.P() = 1, and for {Ai}iI countable disjoint, P(iIAi) =</p><p>iIP(Ai).</p><p>If p is a property with P({w : p(w) is true}) = 1, we say p holds almost surely.</p><p>Example. Rolling a 6-sided die. = {1, 2, . . . , 6}. F = 2</p><p>. P(E) = |E</p><p>|6 .</p><p>Example. Flip a coin. = {H, T}. F = 2. P(H) = p, P(T) = 1 p.</p><p>Example. Random point in [0, 1]. = [0, 1]. F = B[0,1], B = (T) in a topological space(X, T). P = m = Lebesgue measure.</p><p>Immeasurable set example in [0, 1]: on [0, 1] by x y if x y Q. This partitions [0, 1]into (uncountably many) equivalence classes [x], [x], . . . .</p><p>Let V have one representative from each class (using AOC). Let Vr = r + V (mod 1).[0, 1] = rQ = 1 =</p><p>rQ m(Vr).</p><p>Example. Standard normal distribution. = R. F = B. P(E) = 12</p><p>E</p><p>ex2</p><p>2 dx.</p><p>Example. Poisson (). = N0. F = 2. P(E) =</p><p>kEe k</p><p>k!.</p></li><li><p>7/30/2019 Math 6710 Lec 01</p><p> 2/2</p><p>Example. countable, w : [0, 1] w/</p><p>w w() = 1. f : Rn [0, 1] w/</p><p>f(x)dx = 1.</p><p>f is a pdf, so f = dPdm</p><p>whereas w is a pmf for P, so w = dPdc</p><p>.</p><p>A measurable function X : (, F) (S, G) is a (S, G)-valued R.V.</p><p>E.g. X() = 1E() = 1 if E, 0 if Ec.</p><p>If (, F, P) is a prob space and X is a (S, G)-valued R.V, X induces measure m = P X1</p><p>on (S, G). m(A) = P({ : X() A), e.g. P(X A).</p><p>IfX : (S, G) induces -alg on : {X1(E) : E G} = (X), the sigma algebra inducedby X. (X) F.</p><p>Example. In rolling a die, one sigma algebra is {, {1, 3, 5}, {2, 4, 6}, }.</p><p>E[X] =</p><p>XdP (when well-defined)</p><p>Prob/Measure table:</p><p>Probability Measure</p><p> XF P </p><p>X f</p><p>E</p><p>Extensions of probability spaces.</p><p>We say that (, F, P) extends (, F, P) if there exists a measurable surjection : which is probability-preserving, meaning P(1(E)) = P(E).</p><p>E.g. can extend die roll to = {1, 2, . . . , 6}2, F = 2</p><p>, P(E) = E36 , (x, y) = x.</p><p>Probability can only study concepts and perform operations that are preserved by extentions.</p><p>Equality/Non-equality of events is probabilistic. E = F = 1(E) = 1(F). 1(E F) = 1(E) 1(F), etc., so Boolean ops are probabilistic.</p><p>Cardinality is not probabilistic, i.e. not preserved by extension. Completeness isnt either.([0, 1], B, m) can extend to ([0, 1], L, m).</p><p>Page 2</p></li></ul>

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