Q: What does the Central Limit Theorem (CLT) assert?

If {X_n} are i.i.d. with mean μ and variance σ² ∈ (0,∞), then √n((1/n)Σ_{i=1}^n X_i - μ) →^D N(0, σ²) as n→∞, where →^D denotes convergence in distribution (weak convergence).

Q: How do almost sure convergence, convergence in probability, and convergence in distribution relate in strength?

Ordering from strongest to weakest: almost sure convergence ⟹ convergence in probability ⟹ convergence in distribution. Convergence in distribution is the weakest; the converse implications do not hold in general.

Question 1

What is a sample space in probability?

Accepted Answer

The set of all possible outcomes of a random experiment, denoted Ω. Every outcome in the experiment must belong to exactly one element of the sample space.

Question 2

Define a σ-algebra (sigma-algebra) on a set Ω.

Accepted Answer

A collection F of subsets of Ω such that: (1) Ω ∈ F, (2) if A ∈ F then A^c ∈ F, and (3) if {A_n} is a countable collection in F, then ∪A_n ∈ F. Closed under complements and countable unions.

Question 3

What is a probability measure?

Accepted Answer

A function P: F → [0,1] on a σ-algebra F such that: (1) P(Ω) = 1, (2) P(∅) = 0, and (3) for disjoint countable collections {A_n}, P(∪A_n) = ΣP(A_n). Satisfies countable additivity.

Question 4

What is a random variable?

Accepted Answer

A measurable function X: (Ω, F) → (ℝ, B) where B is the Borel σ-algebra on ℝ. Formally, X^{-1}(B) ∈ F for all B ∈ B, ensuring the preimage of measurable sets is F-measurable.

Question 5

Define the cumulative distribution function (CDF) of a random variable X.

Accepted Answer

F_X(x) = P(X ≤ x) for x ∈ ℝ. Properties: non-decreasing, right-continuous, lim_{x→-∞} F_X(x) = 0, lim_{x→∞} F_X(x) = 1.

Question 6

What is the difference between a PDF and a CDF?

Accepted Answer

A PDF f_X(x) satisfies ∫f_X(x)dx = 1 and describes probability density (probability per unit length). A CDF F_X(x) = ∫_{-∞}^x f_X(t)dt is cumulative probability up to x. F'_X(x) = f_X(x) where the derivative exists.

Question 7

Define convergence in probability for a sequence of random variables {X_n}.

Accepted Answer

X_n → X in probability (X_n →^P X) if for all ε > 0: lim_{n→∞} P(|X_n - X| > ε) = 0. Denoted X_n →^P X or plim X_n = X.

Question 8

State the Law of Large Numbers (LLN) and identify which convergence mode it uses.

Accepted Answer

Strong LLN: If {X_n} are i.i.d. with E[|X_i|] < ∞, then (1/n)Σ_{i=1}^n X_i → E[X] almost surely as n→∞. Uses almost sure convergence (stronger than convergence in probability).

Question 9

What does the Central Limit Theorem (CLT) assert?

Accepted Answer

If {X_n} are i.i.d. with mean μ and variance σ² ∈ (0,∞), then √n((1/n)Σ_{i=1}^n X_i - μ) →^D N(0, σ²) as n→∞, where →^D denotes convergence in distribution (weak convergence).

Question 10

How do almost sure convergence, convergence in probability, and convergence in distribution relate in strength?

Accepted Answer

Ordering from strongest to weakest: almost sure convergence ⟹ convergence in probability ⟹ convergence in distribution. Convergence in distribution is the weakest; the converse implications do not hold in general.

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