Cumulative distribution function

Exponential distribution cdf

Normal Distribution CDF

Discrete probability distribution illustration

Discrete probability distribution with a countable set of discontinuities

CDF plot with two red rectangles, illustrating (-x)F(x) and x(1-F(x))

Folded-cumulative-distribution-function

Cumulative distribution function (CDF), in probability theory and statistics, is a function that describes the probability that a real-valued random variable X will take a value less than or equal to x. It is a fundamental concept in the field of probability distributions and is used to specify the distribution of a random variable.

Definition

The cumulative distribution function, \(F(x)\), of a random variable \(X\) is defined as:

\[F(x) = P(X \leq x)\]

where \(P\) denotes the probability that the random variable \(X\) takes on a value less than or equal to \(x\).

Properties

The CDF has several important properties:

• It is a non-decreasing function.
• It is right-continuous.
• \(F(-\infty) = 0\) and \(F(\infty) = 1\), meaning it starts at 0 and asymptotically approaches 1 as \(x\) increases.
• The difference between the CDF values at two points gives the probability of the random variable falling within that interval.

Types of Distributions

There are many types of probability distributions, each with its own CDF. Some common distributions include:

• Normal distribution: Also known as the Gaussian distribution, characterized by its bell-shaped curve.
• Binomial distribution: Describes the number of successes in a fixed number of independent Bernoulli trials.
• Poisson distribution: A discrete frequency distribution which gives the probability of a number of events occurring in a fixed interval of time or space.
• Exponential distribution: Describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Applications

The CDF is widely used in various fields such as engineering, finance, and medicine to model and analyze real-world phenomena. It is essential in determining probabilities, conducting hypothesis tests, and estimating parameters of distributions.

Calculating the CDF

The method of calculating the CDF depends on the type of the distribution (whether it is discrete or continuous) and its parameters. For discrete distributions, the CDF is the sum of the probabilities of the outcomes up to and including \(x\). For continuous distributions, the CDF is the integral of the probability density function (PDF) up to \(x\).

See Also

• Probability density function
• Survival function
• Quantile function

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