The exponential distribution is a continuous probability distribution defined on the positive reals, parameterized by a rate parameter, with probability density function proportional to a decaying exponential and a corresponding cumulative distribution function derivable by direct integration; scaling by the rate parameter reduces any exponential distribution to the standard exponential(1) case, from which the mean (1/rate) and variance (1/rate²) are obtained via direct integration and the law of the unconscious statistician. Its defining theoretical property is memorylessness—the conditional probability of waiting an additional amount of time, given that some time has already elapsed without an event, equals the unconditional probability of waiting that same additional amount—and the exponential distribution is the unique continuous distribution possessing this property, with direct implications for conditional expectation and applications in survival analysis.
The exponential distribution is a continuous probability distribution defined on the positive reals, parameterized by a rate parameter, with probability density function proportional to a decaying ex…