Activity 15
MATH 216: Statistical Thinking
Exponential Distribution: Waiting Time Analysis
Time Allocation: 15 minutes total (5 min reading, 10 min individual work)
Part 1: Conceptual Understanding (3 minutes)
Instructions: Answer the following questions about exponential distributions:
- What does the memoryless property mean in the context of waiting times?
- Why is the exponential distribution particularly suited for modeling waiting times between random events?
- How does the parameter θ relate to both the mean and standard deviation of an exponential distribution?
Part 2: Basic Probability Calculations (4 minutes)
Instructions: Calculate the following probabilities for an exponential distribution with θ = 2:
\(P(X > 3)\) =
\(P(X \leq 1)\) =
\(P(1 < X < 4)\) =
Show your work for at least two calculations:
Part 3: Real-World Applications (8 minutes)
Case 1: Battery Reliability Analysis
Scenario: A company is testing a new battery with an expected lifetime modeled by an exponential distribution with \(\theta = 5\) years. The company wants to offer a warranty period such that only 10% of the batteries fail within the warranty period.
Calculate:
- Warranty period for 90% reliability: years
- Probability battery lasts more than 8 years:
- Median battery lifetime: years
Critical Thinking: What are the business implications of choosing this warranty period?
Case 2: Emergency Room Planning
Scenario: The length of time (in hours) between emergency arrivals at a hospital is modeled as an exponential distribution with \(\theta = 2\) hours.
Calculate:
- Probability of more than 5 hours between arrivals:
- Time by which 75% of arrivals occur: hours
- Probability of arrival within 30 minutes:
Staffing Decision: Based on these probabilities, would you recommend having staff on standby? Why?
Case 3: Quality Control and Warranties
Scenario: A microwave manufacturer tests magnetron tubes with exponential lifetime distribution (\(\theta = 6.25\) years).
Calculate:
- Mean lifetime: years
- Standard deviation: years
- Fraction failing in 5-year warranty:
- Warranty for 27.5% failure rate: years
Business Strategy: What warranty period would you recommend and why?