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3.1 Continuous Random Variables - Introduction

Overview

This section introduces the concept of continuous random variables, building on the discrete probability distributions discussed previously. The session will cover fundamentals and applications of continuous distributions such as the uniform, exponential, and normal distributions.

Key Concepts

  • Continuous Probability Distributions: Contrasting with discrete variables, continuous variables can represent any value within a range, including fractions.
  • Probability Density Function (PDF): Unlike discrete variables where probabilities are defined at specific points, for continuous variables, probabilities are defined over intervals.
  • Cumulative Distribution Function (CDF): Represents the probability that the variable is less than or equal to a certain value.
  • Expected Value and Variance: The concepts are extended from discrete to continuous variables, focusing on integration over summation.

Detailed Explanation

Introduction to Continuous Variables

  • Examples:
  • Time between customer arrivals at a bank.
  • Fluid amount in a bottle labeled 750 ml might actually be between 740 ml to 760 ml.
  • Illustration Recommended: Visualize different continuous variables used in examples with a table or simple diagrams indicating ranges.

Probability Density Function (PDF)

  • Explains how probabilities are not directly associated with specific values but with intervals.
  • Diagram Recommended: Show a PDF graph for a simple continuous distribution (e.g., uniform distribution) and highlight how areas under the curve represent probabilities.

Cumulative Distribution Function (CDF)

  • Describes how CDF is used to calculate the probability of observing a value within a particular range.
  • Diagram Recommended: Include a plot of CDF showing the increasing probability as the variable value increases.

Comparison with Discrete Variables

  • Highlight the differences in calculating probabilities, expected values, and variance between discrete and continuous random variables.
  • Table Recommended: Compare attributes of discrete and continuous variables like value type, PDF/CDF application, and calculation methods for expected value and variance.

Applications and Implications

  • Practical Applications: Discuss practical applications of continuous distributions in business analytics.
  • Example Recommended: Describe a scenario where normal distribution might be applied to model customer behavior or product life expectancy.

Advanced Distributions

  • Briefly introduce other continuous distributions related to the normal distribution:
  • Chi-square
  • T-distribution
  • F-distribution
  • Diagram Recommended: Provide graphs of these distributions to illustrate their shapes and discuss their relevance.
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