Lecture on Poisson Distribution Problems

Introduction

• Discussed how to use the Poisson distribution formula to solve problems.
• Focused on probability problems relating to a time interval.

Problem 1: Customers Per Day

• Scenario: Small business with an average of 12 customers per day.
• Objective: Probability of receiving exactly 8 customers in one day.
• Mean (λ): 12 customers/day.
• Random Variable (X): 8.
• Formula: ( P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} ).
• Calculation:
  • ( \frac{12^8 e^{-12}}{8!} \approx 0.065523 ).
  • Approx. 6.55% probability.

Problem 2: Text Messages in a Time Period

Part A

• Scenario: Student receives 7 messages on average per 2-hour period.
• Objective: Probability of receiving exactly 9 messages in 2 hours.
• Mean (λ): 7.
• Random Variable (X): 9.
• Calculation:
  • ( \frac{7^9 e^{-7}}{9!} \approx 0.1014 ).
  • Approx. 10.14% probability.

Part B

• Scenario: Probability of receiving 24 messages in an 8-hour period.
• Mean (λ): Recalculate for 8 hours. New mean is 28.
• Random Variable (X): 24.
• Calculation:
  • ( \frac{28^{24} e^{-28}}{24!} \approx 0.060095 ).
  • Approx. 6.01% probability.

Problem 3: Calls Per Hour

Part A

• Scenario: Business receives 8 calls per hour on average.
• Objective: Probability of receiving exactly 7 calls in 1 hour.
• Mean (λ): 8.
• Random Variable (X): 7.
• Calculation:
  • ( \frac{8^7 e^{-8}}{7!} \approx 0.1395865 ).
  • Approx. 13.96% probability.

Part B

• Objective: Probability of receiving at most 5 calls in 1 hour.
• Sum of Probabilities: Calculate cumulative probability for X = 0 to 5.
• Calculation:
  • ( e^{-8} \sum_{x=0}^{5} \frac{8^x}{x!} \approx 0.191236 ).
  • Approx. 19.12% probability._

Part C

• Objective: Probability of receiving more than 6 calls in 1 hour.
• Relation with Cumulative Probability: ( P(X > 6) = 1 - P(X \leq 6) ).
• Calculation:
  • ( 1 - \left( e^{-8} \sum_{x=0}^{6} \frac{8^x}{x!} \right) \approx 0.686626 ).
  • Approx. 68.7% probability._

Conclusion

• Importance of understanding the Poisson distribution.
• Techniques for computing probabilities using the Poisson formula.