eksponentiaaljaotus

Eksponentiaaljaotus, also known as the exponential distribution, is a continuous probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is widely used in various fields such as reliability engineering, queuing theory, and survival analysis.

The probability density function (PDF) of the exponential distribution is given by:

f(x; λ) = λ * e^(-λx), for x ≥ 0

where λ (lambda) is the rate parameter, which is the inverse of the mean (1/μ). The cumulative distribution function (CDF) is:

F(x; λ) = 1 - e^(-λx), for x ≥ 0

Key properties of the exponential distribution include:

1. Memorylessness: The exponential distribution is memoryless, meaning that the probability of an event occurring in the future is independent of how much time has already passed. This property makes it useful in modeling systems where the age of the system does not affect the probability of failure.

2. Mean and variance: The mean of the exponential distribution is 1/λ, and the variance is also 1/λ^2.

3. Relationship to the Poisson process: The exponential distribution is closely related to the Poisson process, as the time between events in a Poisson process follows an exponential distribution.

The exponential distribution is often used as a model for the time until the next event in a Poisson process, such as the time between arrivals in a queue or the time until a component fails in a reliability system. However, it is important to note that the exponential distribution assumes that the rate of events is constant over time, which may not always be the case in real-world applications. In such cases, other distributions, such as the Weibull distribution, may be more appropriate.