The exponential distribution is a probability model for waiting times, such as the time until the next bus, radioactive decay event, or customer arrival. It is most useful when events happen randomly but at a constant average rate. In statistics, it is closely connected to the Poisson process, where the number of events in a time interval follows a Poisson distribution.
Understanding this distribution helps students connect rates, probabilities, and real-world waiting times.
Key Facts
- Probability density function: f(t) = λe^(-λt) for t >= 0
- Cumulative distribution function: P(T <= t) = 1 - e^(-λt)
- Survival probability: P(T > t) = e^(-λt)
- Mean waiting time: E(T) = 1/λ
- Variance: Var(T) = 1/λ^2
- Memoryless property: P(T > s + t | T > s) = P(T > t)
Vocabulary
- Exponential distribution
- A continuous probability distribution that models the waiting time until the next event in a Poisson process.
- Rate parameter
- The value λ that represents the average number of events per unit time.
- Probability density
- A function whose area over an interval gives the probability that a continuous random variable falls in that interval.
- Poisson process
- A random process in which events occur independently at a constant average rate over time or space.
- Memoryless property
- The property that the future waiting time has the same distribution no matter how long you have already waited.
Common Mistakes to Avoid
- Using λ as the mean, which is wrong because the mean waiting time is 1/λ, not λ.
- Treating f(t) as a direct probability at one exact time, which is wrong because probabilities for continuous variables come from areas under the curve.
- Forgetting that t must be nonnegative, which is wrong because the exponential distribution models waiting time and negative waiting time has no meaning.
- Applying the exponential distribution when the event rate changes over time, which is wrong because the basic model assumes a constant rate λ.
Practice Questions
- 1 Customers arrive at a service desk at an average rate of 4 per hour. If the waiting time T until the next customer is exponential, find P(T > 0.5 hours).
- 2 A machine fails according to an exponential distribution with mean lifetime 200 hours. Find λ and calculate P(T <= 50 hours).
- 3 A student has already waited 10 minutes for a randomly arriving bus whose waiting time is modeled by an exponential distribution. Explain why the probability of waiting at least 5 more minutes is the same as it was at the start.