In probability theory, what does the Poisson Distribution formula represent?

The Poisson Distribution formula represents the probability of a given number of events happening in a fixed interval of time or space under the assumption that these events occur with a known constant mean rate and independently of the time since the last event. The correct formula, represented as ( p(x) = \frac{\lambda^x e^{-\lambda}}{x!} ), where ( \lambda ) is the average rate of occurrence over the specified interval, ( x ) is the actual number of events, and ( e ) is the base of the natural logarithm, encapsulates these core elements of the distribution.

This distribution is typically used in scenarios where events happen randomly but at a predictable average rate, such as the number of phone calls received at a call center in an hour or the number of emails received in a day, therefore making it particularly useful in fields involving risk management and statistical modeling.

The other choices do not represent the Poisson Distribution: the first choice corresponds to the binomial distribution, which models the number of successful outcomes in a fixed number of trials with two possible outcomes; the second choice refers to a formula used for measuring tracking error in finance, which is not related to the Poisson distribution at all; and the