Class+Notes+2+Probabiliity+and+Probability+Distributions

ECON 130-01: Econometrics Notes

Topic Overview

  • Key Topics:

    • Probability

    • Complement probability, unions, and intersections

    • Conditional probability

    • Statistical independence

    • Multiplicative Rule

    • Discrete Random Variables

    • Continuous Random Variables

    • Uniform Distribution

    • Normal Distribution

Probability Concepts

Definitions

  • Experiment: Any manipulation or observation of the world (e.g., flipping a coin).

  • Sample Point: The most basic outcome of an experiment. For instance, heads or tails in a coin flip.

  • Sample Space: Set of all possible outcomes (e.g., S = {H, T} for a coin flip).

Events

  • Event: A collection of sample points.

  • Complement of an Event (A^C): Outcomes not in event A (e.g., event A is getting at least 1 head in two coin flips).

    • Sample Space Example: S = {HH, HT, TH, TT}.

  • Intersection (A ∩ B): Both events occur.

  • Union (A ∪ B): Either one or both events occur.

Types of Events

  • Exhaustive Events: All possible outcomes.

  • Mutually Exclusive Events: Events that cannot occur together.

  • Equally Likely Events: Events with no preference.

  • Independent Events: One event's occurrence does not affect another (e.g., coin flips).

Probability Assignments

  • Classical Method (A priori): Determining probability based on prior knowledge.

    • Example: Probability of heads in a fair coin flip is 0.5.

  • Empirical Method: Collecting data from repeated experiments.

    • Example: Calculating probability after rolling a die multiple times.

  • Subjective Method: Based on personal estimations and non-quantifiable knowledge (e.g., predicting terrorist attacks).

Compound Events and Probabilities

  • Notation: Pr(A or B) = Pr(A ∪ B) and Pr(A and B) = Pr(A ∩ B).

  • Axioms:

    • P(S) = 1 (total probability).

    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Random Variables

Discrete Random Variables

  • Can take only a finite number of values.

  • Probability Distribution: Lists pairs [x, P(x)], summing to 1.

  • Expected Value (E(X)): Mean of the distribution.

  • Variance (V^2): Measures dispersion around the mean.

Continuous Random Variables

  • Can take on infinite values (e.g., running speed).

  • Discussed via Probability Density Function (f(x)).

    • Area under the curve equals 1.

Distributions

Uniform Distribution

  • All outcomes have equal probability (f(x) = 1/(d-c) for values between c and d).

Normal Distribution

  • Bell-shaped curve; mean, median, and mode are equal.

  • Probabilities are found using z-scores.

  • Standard normal distribution: P = 0, V = 1.

  • Chebyshev’s Inequality: Provides probability estimates for any distribution, applicable beyond normal distributions.