Exponential Distribution Calculator
Calculate probabilities and statistics for exponential distribution with rate parameter
Calculate Exponential Distribution
Average rate of events per time unit (λ > 0)
Time period for probability calculations
Example: Bus Stop Waiting Time
Problem
Scenario: Buses arrive at a stop every 15 minutes on average
Question: What's the probability of waiting less than 3 minutes?
Rate parameter: λ = 1/15 = 0.0667 buses per minute
Time value: X = 3 minutes
Solution
P(x ≤ 3) = 1 - e^(-0.0667 × 3)
P(x ≤ 3) = 1 - e^(-0.2)
P(x ≤ 3) = 1 - 0.8187
P(x ≤ 3) = 0.1813 = 18.13%
There's an 18.13% chance of waiting less than 3 minutes
Key Properties
Memoryless
Future probability independent of past
Continuous
Models continuous time intervals
Skewed
Right-skewed distribution
Common Applications
Waiting times between events
Time until machine failure
Customer service times
Time between accidents
Radioactive decay intervals
Statistical Tips
Rate parameter λ must be positive
Mean = Standard Deviation = 1/λ
Higher λ means shorter waiting times
P(x > X) + P(x ≤ X) = 1
Understanding Exponential Distribution
What is Exponential Distribution?
The exponential distribution describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It's characterized by the "memoryless" property - the probability of an event occurring in the next time interval is independent of how much time has already elapsed.
Key Characteristics
- •Continuous probability distribution
- •Only one parameter: rate λ (lambda)
- •Domain: x ≥ 0 (non-negative values)
- •Right-skewed with long tail
Mathematical Formulas
Probability Density Function (PDF):
f(x) = λe^(-λx)
Cumulative Distribution Function (CDF):
F(x) = 1 - e^(-λx)
Key Statistics:
- Mean: μ = 1/λ
- Median: ln(2)/λ ≈ 0.693/λ
- Variance: σ² = 1/λ²
- Standard Deviation: σ = 1/λ
Real-World Examples
Customer Service
Time between customer arrivals at a service desk, assuming customers arrive randomly at a constant average rate.
Equipment Reliability
Time until failure of electronic components or machinery, widely used in reliability engineering.
Natural Phenomena
Time between earthquakes, radioactive decay events, or phone calls to a call center.