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Home , Almost surely, Complementary event, Lebesgue measure, Measurable function, Measure (mathematics), Measure space

Theory

”A is neither random nor variable.”

Gian-Carlo Rota, M.I.T..

Florian Herzog

2013 Probability

Probability space A W is a unique triple W = {Ω, F,P }: • Ω is its space •F its σ- of events • P its probability Remarks: (1) The Ω is the of all possible samples or elementary events ω: Ω = {ω | ω ∈ Ω}. (2)The σ-algebra F is the set of all of the considered events A, i.e., of Ω: F = {A | A ⊆ Ω,A ∈ F}. (3) The P assigns a probability P (A) to every event A ∈ F: P : F → [0, 1].

Stochastic Systems, 2013 2 Sample space

The sample space Ω is sometimes called the universe of all samples or possible outcomes ω. Example 1. Sample space • Toss of a coin (with head and tail): Ω = {H,T }. • Two tosses of a coin: Ω = {HH,HT,TH,TT }.

• A cubic die: Ω = {ω1, ω2, ω3, ω4, ω5, ω6}. • The positive : Ω = {1, 2, 3,... }. • The reals: Ω = {ω | ω ∈ R}. Note that the ωs are a mathematical construct and have per se no real or scientific meaning. The ωs in the die example refer to the numbers of dots observed when the die is thrown.

Stochastic Systems, 2013 3 Event

An event A is a of Ω. If the ω of the experiment is in the subset A, then the event A is said to have occurred. The set of all subsets of the sample space are denoted by 2Ω. Example 2. Events • Head in the coin toss: A = {H}.

• Odd number in the roll of a die: A = {ω1, ω3, ω5}. • An smaller than 5: A = {1, 2, 3, 4}, where Ω = {1, 2, 3,... }. • A between 0 and 1: A = [0, 1], where Ω = {ω | ω ∈ R}. We denote the of A by Ac = Ω\A. When it is possible to determine whether an event A has occurred or not, we must also be able to determine whether Ac has occurred or not.

Stochastic Systems, 2013 4 Probability Measure I

Definition 1. Probability measure A probability measure P on the countable sample space Ω is a set

P : F → [0, 1],

satisfying the following conditions • P(Ω) = 1.

• P(ωi) = pi.

• If A1,A2,A3, ... ∈ F are mutually disjoint, then

∞ ∞  [  X P Ai = P (Ai). i=1 i=1

Stochastic Systems, 2013 5 Probability

The story so far:

• Sample space: Ω = {ω1, . . . , ωn}, finite! • Events: F = 2Ω: All subsets of Ω P • Probability: P (ωi) = pi ⇒ P (A ∈ Ω) = pi wi∈A Probability of Kolmogorov (1931) for elementary probability: • P(Ω) = 1. • If A ∈ Ω then P (A) ≥ 0.

• If A1,A2,A3, ... ∈ Ω are mutually disjoint, then

∞ ∞  [  X P Ai = P (Ai). i=1 i=1

Stochastic Systems, 2013 6 Uncountable sample spaces

n Most important uncountable sample space for engineering: R, resp. R . Consider the example Ω = [0, 1], every ω is equally ”likely”. • Obviously, P(ω) = 0. • Intuitively, P([0, a]) = a, basic concept: !

Question: Has every subset of [0, 1] a determinable length? Answer: No! (e.g. Vitali sets, Banach-Tarski paradox)

Question: Is this of importance in practice? Answer: No!

Question: Does it matter for the underlying theory? Answer: A lot!

Stochastic Systems, 2013 7 Fundamental mathematical tools

Not every subset of [0, 1] has a determinable length ⇒ collect the ones with a determinable length in F. Such a mathematical construct, which has additional, desirable properties, is called σ-algebra. Definition 2. σ-algebra A collection F of subsets of Ω is called a σ-algebra on Ω if • Ω ∈ F and ∅ ∈ F (∅ denotes the ) • If A ∈ F then Ω\A = Ac ∈ F: The complementary subset of A is also in Ω S∞ • For all Ai ∈ F: i=1 Ai ∈ F The intuition behind it: collect all events in the σ-algebra F, make sure that by performing countably many elementary set operation (∪, ∩,c ) on elements of F yields again an element in F (closeness). The pair {Ω, F} is called .

Stochastic Systems, 2013 8 Example of σ-algebra

Example 3. σ-algebra of two coin-tosses

• Ω = {HH,HT,TH,TT } = {ω1, ω2, ω3, ω4}

•F min = {∅, Ω} = {∅, {ω1, ω2, ω3, ω4}}.

•F 1 = {∅, {ω1, ω2}, {ω3, ω4}, {ω1, ω2, ω3, ω4}}.

•F max = {∅, {ω1}, {ω2}, {ω3}, {ω4}, {ω1, ω2}, {ω1, ω3}, {ω1, ω4}, {ω2, ω3}, {ω2, ω4}, {ω3, ω4}, {ω1, ω2, ω3}, {ω1, ω2, ω4}, {ω1, ω3, ω4}, {ω2, ω3, ω4}, Ω}.

Stochastic Systems, 2013 9 Generated σ-

The concept of generated σ-algebras is important in .

Definition 3. σ(C): σ-algebra generated by a class C of subsets Let C be a class of subsets of Ω. The σ-algebra generated by C, denoted by σ(C), is the smallest σ-algebra F which includes all elements of C, i.e., C ∈ F.

Identify the different events we can measure of an experiment (denoted by A), we then just work with the σ-algebra generated by A and have avoided all the measure theoretic technicalities.

Stochastic Systems, 2013 10 Borel σ-algebra

The Borel σ-algebra includes all subsets of R which are of interest in practical applications (scientific or engineering). Definition 4. Borel σ-algebra B(R)) The Borel σ-algebra B(R) is the smallest σ-algebra containing all open intervals in R. The sets in B(R) are called Borel sets. The extension to the n multi-dimensional case, B(R ), is straightforward. • (−∞, a), (b, ∞), (−∞, a) ∪ (b, ∞) • [a, b] = (−∞, a) ∪ (b, ∞), S∞ S∞ • (−∞, a] = n=1[a − n, a] and [b, ∞) = n=1[b, b + n], • (a, b] = (−∞, b] ∩ (a, ∞), T∞ 1 1 •{ a} = n=1(a − n, a + n), Sn •{ a1, ··· , an} = k=1 ak.

Stochastic Systems, 2013 11 Measure

Definition 5. Measure Let F be the σ-algebra of Ω and therefore (Ω, F) be a . The map µ : F → [0, ∞] is called a measure on (Ω, F) if µ is countably additive. The measure µ is countably additive (or σ-additive) if µ(∅) = 0 and for every of (F : i ∈ ) in F with F = S F we have i N i∈N i X µ(F ) = µ(Fi). i∈N

If µ is countably additive, it is also additive, meaning for every F,G ∈ F we have µ(F ∪ G) = µ(F ) + µ(G) if and only if F ∩ G = ∅ The triple (Ω, F, µ) is called a measure space.

Stochastic Systems, 2013 12

The measure of length on the straight line is known as the Lebesgue measure. Definition 6. Lebesgue measure on B(R) The Lebesgue measure on B(R), denoted by λ, is defined as the measure on (R, B(R)) which assigns the measure of each to be its length. Examples: • Lebesgue measure of one point: λ({a}) = 0. P∞ • Lebesgue measure of countably many points: λ(A) = i=1 λ({ai}) = 0. • The Lebesgue measure of a set containing uncountably many points: – zero – positive and finite – infinite

Stochastic Systems, 2013 13 Probability Measure

Definition 7. Probability measure A probability measure P on the sample space Ω with σ-algebra F is a P : F → [0, 1], satisfying the following conditions • P(Ω) = 1. • If A ∈ F then P (A) ≥ 0.

• If A1,A2,A3, ... ∈ F are mutually disjoint, then

∞ ∞  [  X P Ai = P (Ai). i=1 i=1

The triple (Ω, F,P ) is called a probability space.

Stochastic Systems, 2013 14 F-measurable functions

Definition 8. F- The function f :Ω → R defined on (Ω, F,P ) is called F-measurable if

−1 f (B) = {ω ∈ Ω: f(ω) ∈ B} ∈ F for all B ∈ B(R),

−1 i.e., the inverse f maps all of the Borel sets B ⊂ R to F. Sometimes it is easier to work with following equivalent condition:

y ∈ R ⇒ {ω ∈ Ω: f(ω) ≤ y} ∈ F

This that once we know the (random) value X(ω) we know which of the events in F have happened. •F = {∅, Ω}: only constant functions are measurable •F = 2Ω: all functions are measurable

Stochastic Systems, 2013 15 F-measurable functions

Ω: Sample space, Ai: Event, F = σ(A1,...,An): σ-algebra of events

X(ω):Ω 7→ R R 6

j X(ω) ∈ ω R A2 A1 t (−∞, a] ∈ B

Y A3 −1 Ω A4 X : B 7→ F

X: random variable, B: Borel σ-algebra

Stochastic Systems, 2013 16 F-measurable functions - Example

Roll of a die: Ω = {ω1, ω2, ω3, ω4, ω5, ω6} We only know, whether an even or and odd number has shown up: F = {∅, {ω1, ω3, ω5}, {ω2, ω4, ω6}, Ω} = σ({ω1, ω3, ω5})

Consider the following random variable:

 1, if ω = ω , ω , ω ; f(ω) = 1 2 3 −1, if ω = ω4, ω5, ω6.

Check measurability with the condition

y ∈ R ⇒ {ω ∈ Ω: f(ω) ≤ y} ∈ F

{ω ∈ Ω: f(ω) ≤ 0} = {ω4, ω5, ω6} ∈/ F ⇒ f is not F-measurable.

Stochastic Systems, 2013 17 Lebesgue I

Definition 9. Lebesgue Integral (Ω, F) a measure space, µ :Ω → R a measure, f :Ω → R is F-measurable.

• If f is a , i.e., f(x) = ci, for all x ∈ Ai, ci ∈ R

n Z X fdµ = ciµ(Ai). Ω i=1

• If f is nonnegative, we can always construct a sequence of simple functions fn with fn(x) ≤ fn+1(x) which converges to f: limn→∞ fn(x) = f(x). With this sequence, the Lebesgue integral is defined by Z Z fdµ = lim fndµ. Ω n→∞ Ω

Stochastic Systems, 2013 18 Lebesgue integral II

Definition 10. Lebesgue Integral (Ω, F) a measure space, µ :Ω → R a measure, f :Ω → R is F-measurable. • If f is an arbitrary, measurable function, we have f = f + − f − with

f +(x) = max(f(x), 0) and f −(x) = max(−f(x), 0),

and then define Z Z Z fdµ = f +dP − f −dP. Ω Ω Ω

R + The integral above may be finite or infinite. It is not defined if Ω f dP R − and Ω f dP are both infinite.

Stochastic Systems, 2013 19 Riemann vs. Lebesgue

The most important concept of the Lebesgue integral is that the of approximate sums (as the ): for Ω = R: f(x) f(x)

6 6

f

- x - x ∆x

Stochastic Systems, 2013 20 Riemann vs. Lebesgue integral

Theorem 1. Riemann-Lebesgue integral equivalence Let f be a bounded and on [x1, x2] except at a countable number of points in [x1, x2]. Then both the Riemann and the Lebesgue integral with Lebesgue measure µ exist and are the same:

Z x2 Z f(x) dx = fdµ. x1 [x1,x2] There are more functions which are Lebesgue integrable than Riemann integra- ble.

Stochastic Systems, 2013 21 Popular example for Riemann vs. Lebesgue

Consider the function

 0, x ∈ ; f(x) = Q 1, x ∈ R\Q.

The Riemann integral Z 1 f(x)dx 0 does not exist, since lower and upper sum do not converge to the same value. However, the Lebesgue integral Z fdλ = 1 [0,1]

does exist, since f(x) is the of x ∈ R\Q.

Stochastic Systems, 2013 22 Random Variable

Definition 11. Random variable A real-valued random variable X is a F-measurable function defined on a probability space (Ω, F,P ) mapping its sample space Ω into the R:

X :Ω → R.

Since X is F-measurable we have X−1 : B → F.

Stochastic Systems, 2013 23 Distribution function

Definition 12. Distribution function The distribution function of a random variable X, defined on a probability space (Ω, F,P ), is defined by:

F (x) = P (X(ω) ≤ x) = P ({ω : X(ω) ≤ x}).

From this the probability measure of the half-open sets in R is

P (a < X ≤ b) = P ({ω : a < X(ω) ≤ b}) = F (b) − F (a).

Stochastic Systems, 2013 24 Density function

Closely related to the distribution function is the density function. Let f : R 7→ be a nonnegative function, satisfying R fdλ = 1. The function f is called R R a density function (with respect to the Lebesgue measure) and the associated probability measure for a random variable X, defined on (Ω, F,P ), is Z P ({ω : ω ∈ A}) = fdλ. A for all A ∈ F.

Stochastic Systems, 2013 25 Important Densities I

• Poisson density or probability function (λ > 0):

λx f(x) = e−λ , x = 0, 1, 2,.... x!

• Univariate Normal density

1 1x−µ f(x) = √ e−2 σ 2πσ

. The normal variable is abbreviated as N (µ, σ). n n×n • Multivariate normal density (x, µ ∈ R ;Σ ∈ R ):

1 1 T −1 f(x) = e−2(x−µ) Σ (x−µ) . p(2π)ndet(Σ)

Stochastic Systems, 2013 26 Important Densities II

1 1 • Univariate student t-density ν degrees of freedom (x, µ ∈ R ; σ ∈ R

ν+1 2 1 Γ( )  1 (x − µ) −2(ν+1) f(x) = √2 1 + ν 2 Γ( 2 ) πνσ ν σ

n • Multivariate student t-density with ν degrees of freedom (x, µ ∈ R ;Σ ∈ n×n R ):

Γ(ν+n)  1 −1(ν+n) f(x) = 2 1 + (x − µ)T Σ−1(x − µ) 2 . ν p n ν Γ( 2 ) (πν) det(Σ)

Stochastic Systems, 2013 27 Important Densities II

• The chi square distribution with degree-of-freedom (dof) n has the following density x n−2 − 2 x 2 e 2 f(x) = n 2Γ( 2 ) which is abreviated as Z ∼ χ2(n) and where Γ denotes the gamma function. • A chi square distributed random variable Y is created by

n X 2 Y = Xi i=1

where X are independent standard normal distributed random variables N (0, 1).

Stochastic Systems, 2013 28 Important Densities III

• A standard student-t distributed random variable Y is generated by

X Y = , qZ ν

where X ∼ N (0, 1) and Z ∼ χ2(ν). • Another important density is the Laplace distribution:

1 |x−µ| p(x) = e− σ 2σ with µ and diffusion σ. The variance of this distribution is given as 2σ2.

Stochastic Systems, 2013 29 Expectation & Variance

Definition 13. Expectation of a random variable The expectation of a random variable X, defined on a probability space (Ω, F,P ), is defined by: Z Z E[X] = XdP = xfdλ. Ω Ω With this definition at hand, it does not matter what the sample Ω is. The calculations for the two familiar cases of a finite Ω and Ω ≡ R with continuous random variables remain the same. Definition 14. Variance of a random variable The variance of a random variable X, defined on a probability space (Ω, F,P ), is defined by: Z var(X) = E[(X − E[X])2] = (X − E[X])2dP = E[X2] − E[X]2. Ω

Stochastic Systems, 2013 30 Normally distributed random variables

The shorthand notation X ∼ N (µ, σ2) for normally distributed random variables with parameters µ and σ is often found in the literature. The following properties are useful when dealing with normally distributed random variables: • If X ∼ N (µ, σ2)) and Y = aX + b, then Y ∼ N (aµ + b, a2σ2)). 2 2 • If X1 ∼ N (µ1, σ1)) and X2 ∼ N (µ2, σ2)) then 2 2 X1 + X2 ∼ N (µ1 + µ2, σ1 + σ2) (if X1 and X2 are independent)

Stochastic Systems, 2013 31 I

From elementary probability theory (Bayes rule):

P (A ∩ B) P (B|A)P (A) P (A|B) = = ,P (B) > 0. P (B) P (B)

A∩ B BA

Ω

E(XI ) E(X|B) = B ,P (B) > 0. P (B)

Stochastic Systems, 2013 32 Conditional Expectation II

General case: (Ω, F,P ) Definition 15. Conditional expectation Let X be a random variable defined on the probability space (Ω, F,P ) with E[|X|] < ∞. Furthermore let G be a sub-σ-algebra of F (G ⊆ F). Then there exists a random variable Y with the following properties: 1. Y is G-measurable. 2. E[|Y |] < ∞. 3. For all sets G in G we have Z Z Y dP = XdP, for all G ∈ G. G G

The random variable Y = E[X|G] is called conditional expectation.

Stochastic Systems, 2013 33 Conditional Expectation: Example

Y is a piecewise linear approximation of X. Y = E[X|G]

X(ω) 6

X Y

- ω G1 G2 G3 G4 G5

For the trivial σ-algebra {∅, Ω}: R Y = E[X|{∅, Ω}] = Ω XdP = E[X].

Stochastic Systems, 2013 34 Conditional Expectation: Properties

• E(E(X|F)) = E(X). • If X is F-measurable, then E(X|F) = X.

• Linearity: E(αX1 + βX2|F) = αE(X1|F) + βE(X2|F). • Positivity: If X ≥ 0 surely, then E(X|F) ≥ 0. • Tower property: If G is a sub-σ-algebra of F, then

E(E(X|F)|G) = E(X|G).

• Taking out what is known: If Z is G-measurable, then

E(ZX|G) = Z · E(X|G).

Stochastic Systems, 2013 35 Summary

• σ-algebra: collection of the events of interest, closed under elementary set operations • Borel σ-algebra: all the events of practical importance in R • Lebesgue measure: defined as the length of an interval • Density: transforms Lebesgue measure in a probability measure • Measurable function: the σ-algebra of the probability space is ”rich” enough • Random variable X: a measurable function X :Ω 7→ R • Expectation, Variance • Conditional expectation is a piecewise linear approximation of the underlying random variable.

Stochastic Systems, 2013 36