Distribution function

Distribution Function refers to a mathematical function that describes the probability that a random variable takes on a value less than or equal to a certain value. It is a fundamental concept in the fields of probability theory and statistics, providing a comprehensive way to describe the distribution of random variables. The distribution function is also known as the cumulative distribution function (CDF).

Definition

Given a random variable \(X\), the distribution function \(F(x)\) is defined for every number \(x\) by the formula:

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

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

Properties

The distribution function has several important properties:

• It is non-decreasing: if \(a \leq b\), then \(F(a) \leq F(b)\).
• It is right-continuous: for every \(x\), \(F(x) = F(x+)\).
• It approaches 0 as \(x\) approaches negative infinity and 1 as \(x\) approaches positive infinity: \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to \infty} F(x) = 1\).

Types of Distributions

There are various types of distribution functions, each describing the distribution of different kinds of random variables. Some common types include:

• Discrete distribution: Used for random variables that take on a countable number of distinct values.
• Continuous distribution: Used for random variables that take on an infinite number of values within an interval.
• Mixed distribution: Combines elements of both discrete and continuous distributions.

Examples

• The Binomial distribution is an example of a discrete distribution, often used to model the number of successes in a fixed number of independent Bernoulli trials.
• The Normal distribution, also known as the Gaussian distribution, is a continuous distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Applications

Distribution functions are used in a wide range of applications, including:

• Assessing probabilities and making predictions about future events based on historical data.
• Statistical hypothesis testing, where the distribution of the test statistic under the null hypothesis is used to determine the p-value.
• In quantitative finance, to model the returns of assets and to calculate the Value at Risk (VaR).

See Also

• Probability density function
• Survival function
• Hazard function