Exponential Distribution
The Exponential Distribution is used to model the time between events in a process that occurs continuously and independently at a constant average rate. It is ideal for answering questions like:
- How long until the next earthquake occurs?
- What is the waiting time for the next customer to arrive?
- How long will a machine operate before it breaks down?
Definition
The time \(X\) until the next event follows an exponential distribution if the probability of occurrence in a short interval is proportional to the length of the interval.
A random variable \(X\) has an exponential distribution with rate parameter \(\lambda > 0\), written: \[ X \sim \text{Exponential}(\lambda) \]
Probability Functions
Probability Density Function (PDF)
\[ f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x}, & x \ge 0 \\ 0, & x < 0 \end{cases} \]
- \(\lambda\) is the rate (e.g. arrivals per hour)
- \(\mu = \frac{1}{\lambda}\) is the mean waiting time
Cumulative Distribution Function (CDF)
\[ F(x; \lambda) = \begin{cases} 1 - e^{-\lambda x}, & x \ge 0 \\ 0, & x < 0 \end{cases} \]
Key Properties
Mean (expected value):
\[ E[X] = \frac{1}{\lambda} \]Variance:
\[ V[X] = \frac{1}{\lambda^2} \]Memoryless Property:
\[ P(X > s + t \mid X > s) = P(X > t) \]
The exponential is the only continuous distribution with this property.
Lifetimes and Reliability
Exponential distributions are widely used to model lifespans of components, times between events, and reliability:
- Time to failure of electronics
- Duration between bus arrivals
- Period between incoming phone calls
Let \(\mu = 5\). Then the probability that a device lasts more than 6 years is: \[ P(X \ge 6) = e^{-6 / 5} \approx 0.3012 \]
Example Calculations
Suppose \(\mu = 5\) years:
- \(P(X \ge 6) = e^{-6/5} = 0.3012\)
- \(P(X \le 4) = 1 - e^{-4/5} = 0.5507\)
- \(P(5 \le X \le 6) = e^{-5/5} - e^{-6/5} = 0.3679 - 0.3012 = 0.0667\)
Review Questions
Q1. A call center receives 5 calls per hour.
- What is the expected time between calls?
- What is the probability that the next call comes within 6 minutes?
Q2. The time between bank customers is exponentially distributed with \(\lambda = 0.25\).
- Find the mean wait time.
- What is the probability the next arrival takes more than 5 minutes?
- What is the probability it takes less than 2 minutes?
Q3. A gas station receives 3 customers/hour.
- What is the expected wait time?
- Compute \(E(T)\) and \(\text{Var}(T)\)
- What’s the chance the next customer arrives in 15 minutes?
- What’s the probability no customer arrives in the next 30 minutes?