Probability - 1.1 Probability Spaces

Question 1

Probability Spaces

What is a random experiment characterised by? Mathematically, how do we denote a probability space?

Answer 1

1. The possible outcomes (Ω)
2. The observable events (F)
3. The assignment of probability to these events (P)

So a probability space has these three components, as is denoted by:
(Ω, F, P)

Question 2

Probability Spaces

Definiton 1.1: What is a Sample space?

Answer 2

denoted as Ω

In measure-theoretic probability, Ω can be abstract, but to ground it in probability theory:

• Let Ω be all outcomes of tossing a coin twice {HH,TT,HT,TH}
• A subset A of omega is a set that contains all outcomes relating to that event e.g. let A be the event of getting at least 1 tails, then A = {TT,HT,TH}

Question 3

What does the mathematical symbol ‘:=’ mean?

Answer 3

The symbol “:=” indicates a definition, rather than an equation

e.g.
Ω := C([0,T];[0, inf))

the sample space Ω is the set of all continuous functions with domain 0 to T and codomain of all non-negative reals

Question 4

Probability Spaces

Definiton 1.2: What is a σ-algebra?

Answer 4

Denoted by F - generally referred to as the set of information revealed to us by the random experiment

1) The sample space is an element of F (kind of like symmetric groups in group theory)

2) if A is in F A^c is in F ( where A^c = Ω/A = {x ∈ Ω: x ∉ A}) e.g. if I have the event that I get a space, there must be an event that I don’t get the spade

3) If A₁, A₂, A₃ are in F, so is the countable union of all A_n –> F again, kind of like symmetric groups in group theory –> this means any finite selection in F (so could just be A₁and A₂
* e.g. you have the probability of getting heads, tails and heads or tails. (all must be possible for a sigma-algebra sense)
* For this to be a sigma-algebra, it must be under countable unions, finite unions (while true) is not applicable for the proper definiton.

Question 5

Examples of F-measurable events? Or alternative ways to define a sigma algebra F that can be derived from its definition?

a to d

Answer 5

a) as the empty set has the complement of the entire sample space (2), which is also 1)
b) follows of from De Morgan’s laws
c) so if A₁….A_n ∈ F by (a) as the interception of A_N+1=A_N+2… = ∅ ∈ F therefore the finite union is equal to the countable union (as N+1 and beyond are the empty set) ∈ F by (3)
d) If A ∈ F –> A^c ∈ F, furthermore if B ∈ F –> B^c ∈ F, thus if A, B^c ∈ F it follows by c) that A n B^c ∈ F. but A n B^c = A/B, thus A/B ∈ F

Question 6

Probability Spaces

What is DeMorgan’s complement of the union and intersection of countable sets?

Answer 6

Question 7

Real Analysis knowledge:

If x₁, x₂….. ∈ R, what is the limsup_{n–> ∞} x_n?

Answer 7

• The limsup_{n–> ∞}x_n and lim_{n–> ∞}x_n are two different things, generally:
  
  • limit = the destination (if the sequence actually goes somewhere and coverges, or in theory tends to infinity)
  • limsup/liminf = the horizon lines (upper and lower envelopes) that always exists when the sequences bounces around, that is the ultimate bounds of the long-run behaviour of a sequences

limsup_{n–> ∞}x_n
= limsup_{n–> ∞} sup{x_n,x_n+1….}
= limsup_{n–> ∞} sup_{m >= n}x_m
= inf_n>=1sup_{m >= n}x_m

Question 8

Real Analysis knowledge:

Generally what are the actionable step you could do to find the limsup_{n–> ∞} x_n?

Answer 8

Question 9

Real Analysis knowledge:

Why is the limsup_{n–> ∞} x_n? considered a decreasing function?

Answer 9

Either:

For each nested sequence of x_n, going forwards as you just consider the tail supremum, then either:

• The imsup_{n–> ∞} x_n is constant as say the entire sequence is bounded by positive infinity
• Or for each consectutive tail supremum (say for an oscillating but decreasing sequences) will get smaller as smaller as n gets large. Thus we are looking for the expected terminal value that the sequences will tend to as n–>inf

Question 10

Real Analysis knowledge:

If x₁, x₂….. ∈ R, what is the liminf_{n–> ∞} x_n?

Answer 10

This is a generally an increasing function as for each consectutive tail T_n the infinmum of the sequence can only get smaller or get larger e.g. think of a n osciliating sequences that tends to the limit 0. each tail the infimum will get increasing larger till it tends to zero

Question 11

Real Analysis knowledge:

Generally what are the actionable step you could do to find the liminf_{n–> ∞} x_n?

Answer 11

Question 12

Probability Spaces:

If A₁,A₂…… ∈ F what does we know about the limsup and liminf?

Answer 12

• limsup –> A good way to think about it as imagine for is increasing k >=n there is a set formed from the union of set A_k. say B_1, B_2,B_3 which are all sets in their own right it follows from point 3 that each B_1….B_n is in F. And my point b) it follows that the intersection of these B_1…B_n set is also in F.

A similar defintion hold for the liminf

Question 13

Probability Spaces:

What is the limsup_{n –> ∞} A_n and how does it relate to real analysis?

Answer 13

A similar defintion holds for the liminf

Question 14

Ptobability Spaces:

What do the limsup_n–>∞A_n and liminf_n–>∞A_n actually represent?

Answer 14

So letting w ∈ limsup_n–>∞A_n = ∩_{n ∈ N}⋃_{k >=n} A_k

Then ∀ n∈N ∃ some k>=n such that w ∈ A_k

liminf –> The number of sets for which w is in not A_n is a finite number, thus w is in A_n at some point
limsup –> There are infinitely number of sets for which w is in A_n, there are infinite A_n

Question 15

Probability Space:

How can we prove these two sets are equivalent?

Answer 15

Question 16

General Knowledge:

What does the mathematical symbol ※ mean?

What does the mathematical symbol # mean?

Answer 16

= the cardinality of the set.

※ = “special notes”, “assumption” or “contradiction reached” (sometimes also written as “⊥” in logic). Lecturer used it as a stylistic way to mark “we;ve reached a contradiction)

Question 17

Probability Spaces:

Show these are equilvalent set.

Answer 17

Question 18

Probability Spaces:

When Ω is finite (or countable) what is the usual choice for a sigma-algebra on Ω?

Answer 18

Also can write the power set of Ω as P(Ω)

e.g. if Ω={1,2,3} then the number of elements in 2^Ω is 2³ = 8 and:

P(Ω) = {{∅}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},{1,2,3}}

Question 19

Probability Spaces:

When Ω is uncountable e.g. Ω = R what can we choose for a sigma-algebra on Ω?

Answer 19

• You can create different σ-algebra based on different generators A, but one normally tried to choose the smallest σ-algebra set that contains the generator of the set while meeting the definiton of a σ-algebra.
• note that A itself is a set of the set of the generator. That is why in the definition A ∈ G but A ⊆ G. (e.g if A = {{1,2}} and G {∅,{1,2},{3},{1,2,3}}, then A ⊆ G makes sense as {1,2} ∈ G but saying A ∈ G –> {{1,2}} ∈ G which isnt the case)
• G is any σ-algebra that contains the generator A

Question 20

Probability Spaces:

show that σ(curly-A) is a σ-algebra?

Answer 20

Question 21

Probability Spaces:

Example:

if Ω=R, where we have O_R = {all open sets in R} what is the Borel σ-algebra?

Answer 21

⁴ Even more generally, if (Ω, τ ) is a topological space, one sets BΩ := σ(τ ), and calls this the Borel σ-algebra on Ω.

Question 22

Probability Spaces:

What is a probability measure?

Answer 22

Alternative definition is that:

Given a measurable Space (Ω, F). a probability measure μ on (Ω, F) is a map μ is mapping μ: F –> [0,inf] such that:

1) μ(∅) = 0, μ(Ω) =1
2) A₁, A₂… ∈ F, such that A_i ∩ A_i=∅ for i≠j

and μ(⋃^inf _i=1 A_i)= Σ^inf_i=1 μ(A_i) –> this is referred to as (σ-additivity)

Question 23

Probability Spaces:

Give an example how we can create a probability measure for a dice roll?

Answer 23

Question 24

Probability Spaces:

What is a Dirac measure for w*?

Answer 24

Note that unless Ω = {ω^}, it is false to say that we are sure that the state of the world will be ω(^); we can only say that the state of the world will be ω* P-almost surely;

• elementary element is just a single outcome of the sample space
• What we are saying that by fixing w^, then if our event contains w^, then the probability is 1. If it does the probability is 0.

So let Ω = {1,2,3,4,5,6} in idea of the Dirac measure let w^=6 say. then P[6]=1. This is a probability measure because:
1. P(Ω) = 1 because w^=6 ∈ Ω
2. P(∅)= 0 because w^ is not ∈ ∅
3. If A₁, A₂ are disjoint then exactly one of them contains w(^That one get probabilityiy 1, the rest get 0 so the sum works out

Thus P is a probaiblity measure

Question 25

Probability Spaces: Briefly, how is the P the probability measure for a discrete uniform distribution?

Answer 25

Question 26

Probability Spaces: How do we create a probability of the measurable space of a dice roll?

Answer 26

Question 27

Probability Spaces: Approach this question from a measure-probabilistic approach?

Answer 27

Given that our Probability measure is P[A] = |A|/|Ω| * 4^5 = for one paricular run of a fix rank (say 5,6,7,8,9) then for each of the 5 cards, you can choose nay of 4 suits independently. * 4^5-4 = we are considering only the of different suits, thus for the 4 set of runs for this fix set in which you have a run of all the same suit, we need to remove these from the count *10(4^5-4) = for a fixed run, there are 4^5-4 possible occurances of that run based on the suits. There are 10 sets of possible number cards runs from ace,2,3,4,5 to 10,jack,queen,king * (52 5) = The possible number of picking 5 distinct cards from a set of 5 with **no ordering**

Question 28

Probability Spaces: (a) Let (Ω, F,P) be a probability space. If A ∈ F, then how can we calculate the probability of P(A^c)? (b) if A ⊂ B ∈ F then how can we rewrite P(B) in terms of P(A) and what can we infer about the size of their respective probabilities (c) if A,B ∈ F then what is the formula for P(A∪B) and what can we infer about it size relative to P(A)+P(B) (d) If A₁ ⊂ A₂ ⊂ .... ∈ F then what is P( ⋃_n-->∞ A_n)? (e) If A₁ ⊃ A₂ ⊃ .... ∈ F then what is P( ∩_n-->∞ A_n)? (f) If A₁, A₂ .... ∈ F then what is P( ⋃_n-->∞ A_n) less than or equal to?

Answer 28

Question 29

Probability Spaces: What is the proof of (a)?

Answer 29

Question 30

Probability Spaces: What is the proof of (b)?

Answer 30

Question 31

Probability Spaces: What is the proof of (c)?

Answer 31

Question 32

Probability Spaces: What is the proof of (d)?

Answer 32

Question 33

Probability Spaces: What is the proof of (e)?

Answer 33

Question 34

Probability Spaces: What is the proof of (f)?

Answer 34

Question 35

Probability Spaces: Set intersection basic properties?

Answer 35

Question 36

Probability Spaces: What can we not assume about the set A given if P(A)=0 or 1?

Answer 36

Question 37

What do we mean by P(A) given A ⊂ Ω? consider tossing a coin twice so Ω = {HH,TT,HT,TH} and A= event of getting at least 1 tails

Answer 37

* The probability that A "happens means that the experiment resulted in one of the outcomes that lie in the set A * P(A)= 3/4 as there are three things in set A out of 4 things that are equally likely to occur

Question 38

Answer 38

Question 39

What are some of the generic sigma-fields/sigma-algebras?

Answer 39