math 6710 lec 01
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Sean LiMath 6710 Notes Fall 2013
Grad Probability I
Lecture 1 8/28/13
Course Info
John Pike, [email protected]
OH: MLT 580, WF: 1:30 - 2:30
Website: http://www.math.cornell.edu/~web6710/
Grading: All homework.
Probability is a measure space (, F, P) with P() = 1.
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) =
iIP(Ai).
If p is a property with P({w : p(w) is true}) = 1, we say p holds almost surely.
Example. Rolling a 6-sided die. = {1, 2, . . . , 6}. F = 2
. P(E) = |E
|6 .
Example. Flip a coin. = {H, T}. F = 2. P(H) = p, P(T) = 1 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.
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], . . . .
Let V have one representative from each class (using AOC). Let Vr = r + V (mod 1).[0, 1] = rQ = 1 =
rQ m(Vr).
Example. Standard normal distribution. = R. F = B. P(E) = 12
E
ex2
2 dx.
Example. Poisson (). = N0. F = 2. P(E) =
kEe k
k!.
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Example. countable, w : [0, 1] w/
w w() = 1. f : Rn [0, 1] w/
f(x)dx = 1.
f is a pdf, so f = dPdm
whereas w is a pmf for P, so w = dPdc
.
A measurable function X : (, F) (S, G) is a (S, G)-valued R.V.
E.g. X() = 1E() = 1 if E, 0 if Ec.
If (, F, P) is a prob space and X is a (S, G)-valued R.V, X induces measure m = P X1
on (S, G). m(A) = P({ : X() A), e.g. P(X A).
IfX : (S, G) induces -alg on : {X1(E) : E G} = (X), the sigma algebra inducedby X. (X) F.
Example. In rolling a die, one sigma algebra is {, {1, 3, 5}, {2, 4, 6}, }.
E[X] =
XdP (when well-defined)
Prob/Measure table:
Probability Measure
XF P
X f
E
Extensions of probability spaces.
We say that (, F, P) extends (, F, P) if there exists a measurable surjection : which is probability-preserving, meaning P(1(E)) = P(E).
E.g. can extend die roll to = {1, 2, . . . , 6}2, F = 2
, P(E) = E36 , (x, y) = x.
Probability can only study concepts and perform operations that are preserved by extentions.
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.
Cardinality is not probabilistic, i.e. not preserved by extension. Completeness isnt either.([0, 1], B, m) can extend to ([0, 1], L, m).
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