Lecture Notes: Probability Distributions and Binomial Distribution

Overview

• This lecture is the second in a series on distributions. It follows an initial lecture on binomial distributions.
• Focus: Probability distributions, their characteristics, and applications.

Key Concepts

Binomial Distributions

• Used to determine the probability of two mutually exclusive outcomes.
• Histogram shape changes based on:
  • Number of sampling events
  • Probabilities of events occurring
• Binomial distributions follow basic probability rules:
  • Probability values between 0 and 1
  • Sum of probabilities equals 1
  • Independent events: One event's occurrence does not affect another

Pilot Studies

• Often used when probabilities (P or Q) are unknown prior to data collection.
• Helps make predictions about future probabilities based on collected data.

Example: Red Snapper Fishery

Problem Setup

• NOAA wants to adjust the red snapper fishery size limit in the Gulf of Mexico.
• Current size limit: 40 cm, retention rate: 40%
• Goal: Increase retention to 45%

Calculation Method

• Use total frequency histogram to determine probabilities:
  • Determine probability of catching fish of a specific size
  • Calculate by dividing number of fish of a specific size by total fish sampled
• Current size limit probability: 40%
• New retention goal requires increasing fish size classes included under limit

Example Calculations

1. Finding the Probability of a 70 cm Fish:
   • Divide number of 70 cm fish by total fish for probability
2. Size Limit Adjustment:
   • NOAA wants retention from 40% to 45%
   • Calculate necessary fish inclusion for new limit
   • Example: Lowering size class to 35 cm results in 51% retention

Predicting Future Probabilities

• Calculate expected fish size distribution in larger samples:
  • Example: Sampling 1000 instead of 325 fish
  • Use current probabilities to predict outcomes in larger sample sizes

Calculating Mean of Probability Distribution

• Use frequency distribution and probabilities
• Steps:
  1. List events (e.g., fish sizes) and their frequencies
  2. Determine probabilities by dividing each frequency by total observations
  3. Multiply each event size by its probability
  4. Sum results to find mean
• Example: Mean fish size is slightly over 38 cm

Conclusion

• Understanding and applying distributions allows for prediction and probability assessment in scientific studies.
• Further reading: Section 5.7.3 and practice problems 5.11 to 5.18.
• Next lecture will cover the normal distribution.

Preparation

• Review section 5.7.3 of the text.
• Solve practice problems 5.11 - 5.18.
• Prepare questions for class discussion.