Exponentialverteilungen

Exponentialverteilungen are a family of continuous probability distributions that describe the time between events in a Poisson process. A Poisson process is a process where events occur continuously and independently at a constant average rate. The exponential distribution is characterized by a single parameter, often denoted by lambda ($\lambda$), which represents the rate parameter. The probability density function (PDF) of an exponential distribution is given by $f(x; \lambda) = \lambda e^{-\lambda x}$ for $x \geq 0$, and 0 for $x < 0$. The cumulative distribution function (CDF) is $F(x; \lambda) = 1 - e^{-\lambda x}$ for $x \geq 0$.

A key property of the exponential distribution is its memoryless property. This means that the probability of an event occurring in the future is independent of how much time has already passed. In other words, if an event has not yet occurred by time t, the probability that it will occur in the next interval of length $\tau$ is the same as the probability that it will occur in any other interval of length $\tau$, regardless of t. This property makes it useful for modeling phenomena where past occurrences do not influence future ones, such as the lifetime of electronic components or the time until the next customer arrives at a service desk. The mean of an exponential distribution is $1/\lambda$, and its variance is also $1/\lambda^2$.