Cumulative distribution function

A cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to each value. It fully determines the variable’s probability distribution.

Probability measure: A measure assigning probabilities to events, with total probability one. A CDF records the probabilities assigned to events of the form that a variable is at most a given value.
Random variable: A measurable function that assigns a numerical value to each outcome of a random experiment. The CDF describes the distribution of a random variable’s values.
Kolmogorov–Smirnov test: A statistical test comparing empirical distributions with a specified distribution or with each other. Its test statistic is the largest vertical difference between cumulative distribution functions.
Uniform distribution: A probability distribution assigning equal density across an interval, or equal probability across a finite set. On an interval, its CDF rises linearly between the endpoints.
Cumulative hazard function: The integral of a hazard rate over time, measuring accumulated event risk. It accumulates conditional event rates, not the unconditional probability accumulated by a CDF.
Right-continuous function: A function whose limit from the right at each point equals its value there. Every CDF is right-continuous, including at jumps caused by point probabilities.
Cumulative probability: The probability accumulated over outcomes up to a specified threshold. A CDF is precisely the cumulative probability across thresholds.
Empirical distribution function: The CDF that assigns each observation’s value a cumulative fraction of a finite sample. It estimates an unknown population CDF directly from observed data.
Normal distribution: A continuous distribution with a symmetric bell-shaped density determined by its mean and variance. Its CDF is the familiar smooth S-shaped curve, with no elementary closed form.
Probability generating function: A power series encoding the probabilities of a nonnegative integer-valued random variable. It encodes discrete probabilities algebraically rather than as threshold probabilities.

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