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Sean LiMath 6710 Notes Fall 2013

Grad Probability I

Lecture 1 8/28/13

Course Info

John Pike, jpike@cornell.edu

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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