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Calculate probabilities for waiting times, reliability, and Poisson process inter-arrivals.
Mean = 1/λ = 0.5
Enter λ and x, then click Calculate.
Key Properties:
P(X > s+t | X > s) = P(X > t). The probability of waiting another t units is the same regardless of how long you've already waited. This is unique to the exponential (among continuous distributions). Example: if average bulb life is 1000 hours, a 500-hour-old bulb has the same failure probability as a new one.
If events occur as a Poisson process with rate λ, the time between events follows Exp(λ). Poisson counts events in fixed time; exponential measures time between events. They are two views of the same process.
λ = rate (events per unit time). 1/λ = scale = mean time between events. scipy uses scale = 1/λ. So expon(scale=1/λ) or equivalently expon(scale=mean). Always check which parameterization your library uses.
When the failure rate changes over time (use Weibull). When there's a minimum time before an event can occur (use shifted exponential). When events are not independent (use more complex models). When data is bimodal or has heavy tails.
The MLE (maximum likelihood estimator) is λ̂ = n / Σxᵢ = 1/x̄. Simply take the reciprocal of the sample mean. For example, if average time between events is 5 minutes, λ̂ = 1/5 = 0.2 events per minute.