Coursework for Probability for Statistics 2024-2025
Deadline: November 15th, 12PM (noon) on Turnitin
Your solutions should be written in LaTeX. This coursework contains two questions (worth 100 marks total) and a small puzzle at the end for fun (worth 0 marks).
Q1: The Infinite Coin Toss Model (80 Marks)
In this question we think about how to model probabilities for an infinite se- quence of random variables. A natural and simple example is to consider a probability space describing the outcome of infinitely many fair coin tosses.
Let Ω = {H, T}N , i.e. the set of all possible infinite sequences of heads (H) and tails (T). An element ω ∈ Ω will be of the form. ω = (ω1 ,ω2 ,ω3,...), where ωi ∈ {H, T}.
For n ∈ N define
Fn = {[An] for all An ⊂ {H, T}n },
where [An] ⊂ Ω is the set of all infinite coin toss sequences for which the outcome of the first n tosses lies in An , i.e.
[An] = {ω ∈ Ω : (ω1,...,ωn ) ∈ An }.
For example, for A1 = {H}, the set [A1] is the set of all sequences of coin tosses for which the first toss is a H.
1. Show that Fn ⊂ Fm for all m ≥ n. Hint: Do it for m = n + 1 first (10 marks).
2. Prove that F∞ = ∪n∈NFn isan algebra. Hint: First show that [An ∪Bn] =
[An] ∪ [Bn] and [An(c)] = [An]c (20 marks).
3. Is F∞ a σ-algebra? Justify your answer. Hint: Consider the set F = {ω ∈ Ω : every odd toss is H} (20 marks).
4. Let P : F∞ → R be defined by
P([An]) = 2n/|An|,
where |An | denotes the number of elements in An. Show that P defines a probability measure on F∞ (10 marks).
5. Let F = σ(F∞ ) the smallest σ-algebra containing F∞ . Briefly explain how P can be extended to a unique probability measure on F (5 marks).
6. Let A be the event of getting heads infinitely often (i.o.). Show that P(A) = 1. Hint: It might be easier show that P(Ac ) = 0 using continuity of probability measures. (15 marks)
Q2: Lebesgue Measure (20 marks)
Provide a proof for the following propositions.
• The Lebesgue measure of a countable set in R is 0 (10 marks).
• There exists nouncountably additive probability measure on ([0 , 1], B[0 , 1], P) such that P({x1 }) = P({x2 }) for all x1 , x2 ∈ [0, 1]. I.e. it cannot be that for a disjoint sequence of sets At indexed by t ∈ [0, 1]
(10 marks).
Q3: Puzzle (0 Marks)
• True or false. Any nested collection of subsets (call it M) of a countable set is always countable, e.g. if E1 , E2 ∈ M then either E1 ⊂ E2 or E2 ⊂ E1 . If true, give a proof, if false give a counter example.