Exponential Distribution Calculator

The exponential distribution is the standard model for the waiting time until a random event occurs, when those events happen at a steady average rate, and this calculator works out everything you usually need from it. Think of the time between phone calls arriving at a help desk, the gap between buses, the lifetime of certain electronic components, or the interval between arrivals in any queue: when the underlying rate is constant and events are independent, these waiting times follow an exponential distribution. The single thing that controls the whole distribution is the rate parameter, written lambda, which is the average number of events per unit of time. Enter lambda and a value x, and the calculator returns the probability density at x, the cumulative probability that the wait is less than x, the probability that it is greater than x, and the distribution's mean and variance, all updating as you type. The two probabilities are the figures people reach for most: the chance an event happens within a certain time, and the chance you are still waiting after it. They are mirror images that always add to one, because either the event has happened by time x or it has not. A defining quirk of this distribution is that it is memoryless, meaning the chance of waiting another ten minutes is the same whether you have just started waiting or have already waited an hour, which makes it both elegant and slightly counterintuitive. The tool is genuinely useful for statistics students, for reliability and queuing problems in engineering and operations, and for anyone modelling random arrivals or lifetimes. The formulas and a worked example are explained clearly below.

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