Quantitative Economics Notes PDF

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Notes on the Quantitative Economics Module...

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Probability Sample space: the set of possible outcomes (Ω). Elements in Ω will be denoted ω. E.g. Flip two coins: Ω = {(H, H),(H, T),(T, H),(T, T)} Event: a subset of the sample space (A) - These can consist of multiple outcomes E.g. Flipping two coins. A = {(H, H),(H, T)} i.e., event that a head appears on the first coin Bracketing notation: Discrete: Ω = {H, T} Continuous: Ω = [0, ∞) Some Special Events: Elementary Events: each outcome in the sample space Null Event (⦰): the event consisting of no outcomes Certain Event (Ω): the whole sample space Probability Space: (Ω, A, P) Ω : Sample Space A: a nonempty collection of subsets of Ω. Collection of events to which we want to assign probabilities P: is a probability measure on A Definition: A nonempty collection of subsets A of Ω is called σ-field of subsets of Ω provided that - If A is in A, then A c is also in A - If An is A, n = 1, 2, ..., then U∞ n=1An and n∞n=1An (intersect and union) are both in A Definition: A probability measure P on a σ-sigma field of subsets A of a set Ω is a real-valued function having domain A satisfying: - P(Ω) = 1 - P(A) >_ (greater than or equal to) 0 for all A in A - If An, n = 1, 2, 3..., are mutually disjoint sets in A then

Definition: A probability space, denoted (Ω, A, P), is a set Ω, a σ-field of subsets A, and a

probability measure P defined on A Random Variables Discrete r.v.: takes on at most a countable number of possible values Continuous r.v.: takes values on a continuum of possible values CRV Definition: A random variable X on a probability space (X, A, P) is a real-valued function X(ω), ω 3 (member of) Ω, such that for -∞ < x < ∞, {ω|X(ω)...