QNT 2020 - Continuous Probability Distributions (Lecture Note)

Continuous Probability Distributions Overview

  • Course Title: QNT 2020 – Spring 2025 Continuous Probability Distributions

  • Author: Professor Yuan-Mao Kao

  • Chapter Reference: Chapter 7 of the Textbook (Doane & Seward)

Chapter Outline

  • Review probability theory concepts

  • What is a random variable?

  • Discrete variables

  • Discrete probability distributions

  • Continuous Distributions

  • Continuous Probability Distributions

  • Uniform Continuous Distribution

  • Normal Distribution

  • Standard Normal Distribution

  • Normal Approximation

Probability vs. Statistics

  • Statistics involves:

    • Real World application

    • Prediction and Estimation

  • Key distinction:

    • Data (Samples) vs. Model (Distribution)

Data Types Review

  • Caution regarding rounding continuous data to integers

  • Data Types Identified:

    • Qualitative: Verbal label or coded?

    • Quantitative:

      • Discrete: Countable values

      • Continuous: Measurable values

  • Example Data from a Sample of 4,801 U.S. Taxpayers:

    • Variables include: TaxPaid, AGI, Tax %, Filing Type, Child Exemptions

Discrete Random Variables

  • Definition: A random variable assigns a numerical value to each outcome in a random experiment.

  • Notation:

    • Uppercase letters (e.g., 𝑋, π‘Œ) represent random variables.

    • Lowercase letters (e.g., π‘₯, 𝑦) denote specific values of the random variable.

  • Characteristics:

    • Countable number of distinct values

    • Finite sets (e.g., coin tosses) versus infinite countable sets (e.g., trial until heads)

Discrete Probability Distributions

  • Definition: Assigns a probability to each value of a discrete random variable 𝑋.

  • Validity Conditions:

    • Probability of any value within

      • Constraints: 0 ≀ P(X) ≀ 1

      • Total probability must sum to 1

  • Assignments:

    • Multiple sample outcomes can match to a single number, but not vice versa.

    • Multiple random variable values can match the same probability.

Coin Flips Example

  • Random Variable: 𝑋 (number of heads)

  • Definition of possible outcomes from three coin tosses

  • Sum of probabilities must equal 1.

Distribution Functions

  • Probability Distribution Function (PDF):

    • Shows probabilities for each value or interval for discrete variables.

    • Cumulative Distribution Function (CDF):

      • Shows cumulative probabilities summing from smallest to largest values.

    • Key Parameters:

      • Mean, variance, and distribution shape depend on PDF parameters.

Expected Value

  • The expected value (E[X]) is the weighted sum of all X-values by their probabilities.

  • Represents both expectation and mean as a measure of central tendency.

Variance of A Discrete Random Variable

  • Definition: Weighted average of the dispersion of the mean.

  • Notation Variance: 𝜎², V[X].

  • Measure of variability; standard deviation is the square root of variance.

Continuous Random Variables: Events as Intervals

  • Continuous variables defined by probability intervals (e.g., P(X < 54)).

  • Defined as area under a curve (PDF).

Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

  • PDF (f(x)): Non-negative, total area equals 1.

  • CDF (F(x)): Shows cumulative probability up to a certain value.

Understanding Probabilities as Areas

  • Single point probabilities = 0 in continuous case.

  • Total area under PDF = 1.

Expected Value and Variance for Continuous Variables

  • Expected value parallels discrete but uses integrals instead of summation.

  • Variance for continuous calculated similarly to discrete.

Uniform Continuous Distribution

  • Definition: Simplest continuous distribution; PDF constant between limits a, b.

    • Denotated by U(a, b).

  • Example with anesthesia duration (15-30 minutes).

Normal Distribution Characteristics

  • Definition: Bell-shaped, defined by parameters mean (ΞΌ) and standard deviation (Οƒ).

  • Properties:

    • Approximately 99.7% of area within ΞΌ Β± 3Οƒ.

    • Symmetrical and unimodal around the mean.

Standard Normal Distribution

  • A special case of normal where ΞΌ = 0, Οƒ = 1.

  • Transformation allows use of z-tables for cumulative probability calculations.

Empirical Rule for Normal Distribution

  • Key intervals specify how much data lies within standard deviations from the mean:

    • 1Οƒ: 68.26%

    • 2Οƒ: 95.44%

    • 3Οƒ: 99.73%

Finding Areas with Standardized Variables

  • Process to determine student exam scores in a percentile.

  • Empirical calculations and software recommendations for precise results.

Inverse Normal Distribution

  • Technique to find specific percentiles based on cumulative probability.

  • Application to classroom scenarios in normal distributions.