Question

Let (Ω,F,P) be a probability space. Recall the R.V. X is called simple, if there exist A₁, …., Aₙ ∈ F and b₁, …, bₙ ∈ R such that X = ᵢ₌ₗ∑ⁿ bᵢ 1{A ᵢ }. Note that A; need not be disjoint. ...) ... a) Show that if R.V. X,Y are simple, then so is the linear combination aX + bY. b) Is it true that if X₁, X₂ are all simple then so are X₁ + X₂, and X₁ ∧ X₂ = min X1, X₂? c) Is the representation of X (i.e. the sets A₁,.., Aₙ and the scalars b₁, ..., bₙ) unique? d) What if Aᵢ are known to be disjoint? How many values can X take then? e) Show that R.V. X is simple if and only it takes finitely many values. f) Show that E[X] is well defined for X simple R.V.. In other words, if X has two representations: X = ᵢ₌ₗ∑ⁿ bᵢ 1{Aᵢ } and X = ᵢ₌ₗ∑ᵐ Cᵢ 1{Bᵢ), such that B₁, ..., Bₘ ∈ F and C₁, ..., Cₘ ∈ R, then we must have ᵢ₌ₗ∑ⁿ Bᵢ P(Aᵢ) = ᵢ₌ₗ∑ᵐ cᵢP(Bᵢ).

Answer 1

see below

Explanation:

a) If random variables X and Y are simple, then their linear combination aX + bY is also simple.

b) If X₁ and X₂ are simple, then X₁ + X₂ and X₁ ∧ X₂ are also simple.

c) The representation of a simple random variable X is not unique. There can be different sets of {Aᵢ} and corresponding scalars {bᵢ} that represent the same X.

d) If the sets Aᵢ are known to be disjoint, then X can take as many unique values as there are disjoint sets {Aᵢ}.

e) A random variable X is simple if and only if it takes a finite number of values.

f) The expected value E[X] is well-defined for a simple random variable X, regardless of its representation. It can be calculated using the formula E[X] = ∑ᵢ bᵢ P(Aᵢ) or E[X] = ∑ᵢ cᵢ P(Bᵢ), where {Aᵢ} and {Bᵢ} are sets and {bᵢ} and {cᵢ} are scalars representing X.