The exponential distribution is a probability distribution that is commonly used to model the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.
The parameter of the exponential distribution is the rate parameter, denoted as λ. It is the average number of events that occur in a unit time interval. The scale parameter, denoted as β, is often used interchangeably with λ and represents the average time between events.
The parameter of the exponential distribution can be estimated using maximum likelihood estimation (MLE) or method of moments (MOM). MLE involves finding the value of λ that maximizes the likelihood function of the observed data. MOM involves equating the theoretical mean or variance of the exponential distribution to the corresponding sample statistic and solving for the parameter.
The parameter of the exponential distribution determines the shape of the distribution. A smaller value of λ results in a steeper curve and a larger value of λ results in a flatter curve. The scale parameter, β, can also affect the shape of the distribution, with a smaller value resulting in a wider curve and a larger value resulting in a narrower curve.
The parameter, λ, represents the average number of events that occur in a unit time interval. This means that for every unit increase in time, the number of events is expected to increase by a factor of e (Euler's number) to the power of λ. It can also be interpreted as the failure rate, with a higher value indicating a higher failure rate and a lower value indicating a lower failure rate.