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What is a sample space in probability?
The set of all possible outcomes of a random experiment, denoted Ω. Every outcome is mutually exclusive and collectively exhaustive.
Define a σ-algebra (sigma-algebra) on a set Ω.
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.
What is a probability measure?
A function P: F → [0,1] on a σ-algebra F where P(Ω) = 1 and P is countably additive: P(∪A_n) = ΣP(A_n) for disjoint sets A_n.
What is a random variable formally?
A measurable function X: Ω → ℝ such that {ω: X(ω) ≤ x} is in the σ-algebra F for all x ∈ ℝ. It assigns numerical values to outcomes.
Define the cumulative distribution function (CDF) of a random variable X.
F_X(x) = P(X ≤ x) for all x ∈ ℝ. Properties: non-decreasing, right-continuous, lim_{x→-∞} F_X(x) = 0, lim_{x→∞} F_X(x) = 1.
What is the difference between a probability mass function (PMF) and probability density function (PDF)?
PMF applies to discrete random variables: p_X(x) = P(X = x). PDF applies to continuous variables: f_X(x) satisfies P(a ≤ X ≤ b) = ∫_a^b f_X(x)dx and ∫_{-∞}^∞ f_X(x)dx = 1.
Define the expectation (expected value) of a random variable X.
E[X] = ∫_Ω X(ω)dP(ω). For discrete: E[X] = ΣxP(X=x). For continuous: E[X] = ∫_{-∞}^∞ xf_X(x)dx. Represents the center of mass of the distribution.
What is the variance of a random variable and why does it matter?
Var(X) = E[(X - E[X])²] = E[X²] - (E[X])². Measures the spread of a distribution around its mean. Higher variance indicates greater variability.
State the Law of Large Numbers (LLN).
If X₁, X₂, ... are i.i.d. random variables with finite mean μ, then the sample mean (X₁+...+X_n)/n converges in probability to μ as n→∞. Justifies using sample averages to estimate population means.
State the Central Limit Theorem (CLT) and its significance.
If X₁, X₂, ... are i.i.d. with mean μ and variance σ², then √n((X̄_n - μ)/σ) converges in distribution to N(0,1) as n→∞. This explains why normal distribution appears widely and enables inference for large samples.
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