Poisson Distribution and Kaplan-Meier Survival Analysis
What Is the Poisson Distribution?
The Poisson distribution describes the probability of a number of random events occurring in a fixed interval of time or space, such as the number of defects per unit, failures per time period, or pinholes in a given area. Its mean equals its variance, and when the event rate is lambda, the probability of exactly k events is P(k) = lambda^k x e^(-lambda) / k!. The tool accepts lambda and k and computes point and cumulative probabilities directly, including the tail probability of k or more events for risk assessment and sampling judgments.
Kaplan-Meier Survival Analysis
The Kaplan-Meier method is a nonparametric survival analysis that estimates the survival function with the product-limit method and can handle right-censored data (samples that have not failed by the end of the study). It does not assume that failure times follow a particular distribution, making it suitable for exploratory analysis before fitting parametric models such as Weibull or exponential to model the data further. The KM curve estimates survival at each event time in stepwise fashion, and the steps show visually the periods where the failure rate changes.
Applicable Scenarios
Probability prediction for defect-count quality data (surface scratches, solder-joint defects); initial description of lifetime data and median survival time estimation; and handling censored data in reliability verification. The Poisson distribution is also the theoretical basis for defect-count control charts such as the c-chart and u-chart, and the two can be used together. For small counts or many zero-defect batches, Poisson calculations are more accurate than a normal approximation, and for count data such as defects per unit the distribution can be applied directly to control limits and probability judgments.
Outputs and Precautions
The tool outputs the Poisson probability table, cumulative probabilities, survival curve, median survival time and survival rates at each time point. Note that Poisson assumes independent events and a constant rate; survival analysis requires failure times with a censoring flag; and with very small samples the survival-curve steps have large error, so interpret them with confidence intervals. Export the charts and data tables for direct use in analysis reports, and combine the survival curve with confidence intervals for a more reliable lifetime judgment.