Probability 1 CEU Budapest, fall semester 2013 Imre Péter Tóth Homework sheet 1 – due on 23.09.2013

1. Define a σ-algebra as follows:

Definition 1 For a nonempty set Ω, a family F of subsets of ω (i.e. F ⊂ 2^{Ω}, where 2^{Ω} := {A : A ⊂ Ω} is the power set of Ω) is called a σ-algebra over Ω if

• ∅ ∈ F

• if A ∈ F, then A^{C} := Ω \ A ∈ F (that is, F is closed under complement taking)

• if A_{1}, A_{2}, ⋯ ∈ F, then (∪_{i=1}^{∞}A_{i}) ∈ F (that is, F is closed under countable union).

Show from this definition that a σ-algebra is closed under countable intersection, and under finite union and intersection.

2. (a) We toss a biased coin, on which the probability of heads is some 0 ≤ p ≤ 1. Define the random variable ξ as the indicator function of tossing heads, that is

ξ := \begin{cases}0, if tails \\ 1, if heads \end{cases} .

i. Describe the distribution of ξ (called the Bernoulli distribution with parameter

p) in the “classical” way, listing possible values and their probabilities, ii. and also by describing the distirbution as a measure on ℝ, giving the weight

ℙ(ξ ∈ B) of every Borel subset B of ℝ. iii. Calculate the expectation of ξ.

(b) We toss the previous biased coin n times, and denote by X the number of heads tossed.

i. Describe the distribution of X (called the Binomial distribution with parameters

(n, p)) by listing possible values and their probabilities. ii. Calculate the expectation of X by integration (actually summation in this case)

using its distribution, iii. and also by noticing that X = ξ_{1} + ξ_{2} + ⋯ + ξ_{n}, where ξ_{i} is the indicator of the

i-th toss being heads, and using linearity of the expectation.

3. The Fatou lemma is the following

Theorem 1 Let (Ω, F, μ) be a measure space and f_{1}, f_{2}, … a sequence of measureabale functions f_{n} : Ω → ℝ, which are nonneagtive, e.g. f_{n}(x) ≥ 0 for every n = 1, 2, … and every x ∈ Ω. Then

∫_{Ω} lim inf_{n→∞} f_{n}(x) dμ(x) ≤ lim inf_{n→∞} ∫_{Ω} f_{n}(x) dμ(x)

(and both sides make sense).

Show that the inequality in the opposite direction is in general false, by choosing Ω = ℝ, μ as the Lebesgue measure on ℝ, and constructing a sequence of nonnegative f_{n} : ℝ → ℝ for which f_{n}(x) ⟶^{n→∞} 0 for every x ∈ ℝ, but ∫_{ℝ} f_{n}(x) dx ≥ 1 for all n.

4. The ternary number 0.a_{1}a_{2}a_{3} … is the analogue of the usual decimal fraction, but writing numbers in base 3. That is, for any sequence a_{1}, a_{2}, a_{3}, … with a_{n} ∈ {0, 1, 2}, by definition

0.a_{1}a_{2}a_{3} ⋯:= ∑_{n=1}^{∞} \frac{a_{n}}{3^{n}} .

Now let us construct the ternary fraction form of a random real number X via a sequence of fair coin tosses, such that we rule out the digit 1. That is,

a_{n} := \begin{cases}0, if the n-th toss is tails, \\ 2, if the n-th toss is heads \end{cases} ,

and setting X = 0.a_{1}a_{2}a_{3} … (ternary). In this way, X is a “uniformly” chosen random point of the famous middle-third Cantor set C defined as

C := \{∑_{n=1}^{∞} \frac{a_{n}}{3^{n}} , a_{n} ∈ \{0, 2\} (n = 1, 2, …)\} .

Show that

(a) The distribution of X gives zero weight to every point – that is, ℙ(X = x) = 0 for every x ∈ ℝ. (As a consequence, the cumulative distribution function of X is continuous.)

(b) The distribution of X is not absolutely continuous w.r.t the Lebesgue measure on ℝ.