3.5 Hypergeometric and Negative Binomial Distributions

Contents

• 3.5.1 The Hypergeometric Distribution
• 3.5.2 The Negative Binomial Distribution
• EXERCISES Section 3.5 (68—78)

Introduction

Relationships Between Distributions:

• Binomial Distribution:
  • Serves as the approximate probability model for sampling without replacement from a finite dichotomous population (successes $S$ and failures $F$).
  • Applies when:
    • The sample size $n$ is small relative to the population size $N$.
  • Represents the number of successes $X$ when the number of trials $n$ is fixed.
• Hypergeometric Distribution:
  • Provides the exact probability model for the number of successes $S$ in a sample drawn from a finite population.
  • Used when:
    • Sampling without replacement.
  • Both the population size $N$ and the number of successes in the population are known.
• Negative Binomial Distribution:
  • Arises when the number of successes $S$ desired is fixed, while the number of trials becomes random.
  • Focuses on the number of trials needed to achieve a specified number of successes.
• Summary of Differences:
  • Binomial:
    • Fixed number of trials; counts successes.
  • Hypergeometric:
    • Exact count of successes in a fixed sample from a finite population.
  • Negative Binomial:
    • Fixed successes; random trials.
• Conclusion:
  • Understanding these distributions is essential for modeling different sampling scenarios in statistics, particularly in contexts of success/failure outcomes.