Exponential Distribution: Memoryless Property and Survival Models
Understand the exponential distribution, its memoryless property, and survival model applications.
Definition and Basic Properties
If X ~ Exponential(lambda), the PDF is f(x) = lambda * e^(-lambda*x) for x > 0. The CDF is F(x) = 1 - e^(-lambda*x), and the survival function is S(x) = e^(-lambda*x). The mean is 1/lambda, the variance is 1/lambda^2, and the MGF is M(t) = lambda / (lambda - t) for t < lambda.
Some textbooks parameterize with theta = 1/lambda so that f(x) = (1/theta) * e^(-x/theta), with mean theta. Be careful about which parameterization Exam P uses in a given problem.
The Memoryless Property
The exponential distribution is the only continuous distribution with the memoryless property: P(X > s + t | X > s) = P(X > t) for all s, t > 0. In insurance terms, if the time to a claim is exponential, knowing that no claim has occurred in the first 5 years does not change the probability distribution of the remaining waiting time.
This property is frequently tested on Exam P. A typical question provides the memoryless property as a condition and asks you to identify or work with the exponential distribution.
Connections to Other Distributions
The exponential distribution is a special case of the gamma distribution with shape parameter 1. If X1, ..., Xn are independent Exponential(lambda), their sum follows a Gamma(n, lambda) distribution. The minimum of independent exponentials is also exponential: if Xi ~ Exp(lambda_i), then min(X1, ..., Xn) ~ Exp(lambda_1 + ... + lambda_n). This result is useful in competing risks models where the first event to occur determines the outcome.