Introduction
The exponential random variable is a continuous probability distribution that describes the time between independent events occurring at a constant average rate. It is widely used in reliability engineering, queuing theory, survival analysis, and various stochastic processes.
Definition
If a random variable X follows an exponential distribution with parameter λ > 0, we write:
X ~ Exp(λ)
Here, λ is the rate parameter, which represents the average number of events per unit time. The mean time between events is 1/λ.
Probability Density Function (PDF)
f(x) = λ * e-λx for x ≥ 0
f(x) = 0 for x < 0 The PDF shows that the probability decreases exponentially as x increases. The distribution is heavily right-skewed.
Cumulative Distribution Function (CDF)
F(x) = 1 - e-λx for x ≥ 0
F(x) = 0 for x < 0 The CDF gives the probability that the time until the next event is less than or equal to x.
Mean and Variance
- Mean:
E[X] = 1/λ - Variance:
Var(X) = 1/λ²
Key Property: Memorylessness
The exponential distribution is the only continuous distribution that is memoryless. That is:
P(X > s + t | X > s) = P(X > t)
This means the probability that the process lasts at least another t units of time does not depend on how much time has already passed.
Example
Suppose the average number of phone calls received by a call center is 4 per hour. Then the time between two consecutive calls is exponentially distributed with λ = 4.
- Probability that you wait less than 15 minutes (0.25 hours) for the next call:
P(X < 0.25) = 1 - e-λx = 1 - e-4 * 0.25 = 1 - e-1 ≈ 0.632So there's about a 63.2% chance a call comes within the first 15 minutes.
Applications
- Modeling time between arrivals in a Poisson process
- Reliability of systems (e.g. lifespan of electronic components)
- Survival analysis in medical statistics
- Queuing systems (e.g., wait time until next customer)
Python Code Example
from scipy.stats import expon
Set lambda (rate) = 4 => scale = 1/lambda
scale = 1 / 4
Mean and variance
mean, var = expon.mean(scale=scale), expon.var(scale=scale) print("Mean:", mean) print("Variance:", var)
Probability of waiting less than 15 minutes (0.25 hours)
p = expon.cdf(0.25, scale=scale) print("P(X < 0.25):", p)Conclusion
The exponential random variable is essential for modeling the timing of random events. Its simplicity and powerful properties—particularly memorylessness—make it foundational in probability theory, stochastic modeling, and real-world applications involving waiting times and failure rates.