Final answer:
The exponential distribution is characterized by two parameters, the decay parameter (m) and the mean (μ), and is used to model time between events in a Poisson process. It is memoryless, meaning past events don't affect future probabilities. The distribution is related to the Poisson distribution, with the time between events being exponentially distributed.
Step-by-step explanation:
The statement that 'each distribution that is a member of the exponential family has two parameters' is a characterization of a common property in statistics. In particular, the exponential distribution is an example of a continuous distribution that belongs to this family. The exponential distribution is used to model the time between events in a Poisson process and is defined by two parameters: the decay parameter (m) and the mean (μ).
The probability density function (pdf) for the exponential distribution is given by f(x) = me-mx where x ≥ 0 and m > 0. The mean and the standard deviation of an exponential distribution are both equal to 1/m. It is a memoryless distribution, meaning that the probability of an event occurring in the future is independent of any past events. This characteristic is often useful in modeling scenarios like product reliability or the time between random events such as emergency hospital arrivals.
Moreover, the exponential distribution is related to the Poisson distribution. If events occur continuously and independently at a constant average rate, the time between events is exponentially distributed, while the number of events per unit time follows a Poisson distribution with mean λ = 1/μ.