Marginal distribution

Overview of marginal distribution in statistics

Marginal Distribution

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A visualization of a multivariate normal distribution, showing marginal distributions.

In probability theory and statistics, a marginal distribution is the distribution of a subset of a collection of random variables. It provides the probabilities or probability densities of the variables in the subset, without reference to the values of the other variables.

Marginal distributions are derived from the joint probability distribution of the variables. If we have a joint distribution of two random variables, say \(X\) and \(Y\), the marginal distribution of \(X\) is obtained by summing or integrating over all possible values of \(Y\). Similarly, the marginal distribution of \(Y\) is obtained by summing or integrating over all possible values of \(X\).

Mathematical Definition

Consider two random variables \(X\) and \(Y\) with a joint probability distribution \(P(X, Y)\). The marginal probability distribution of \(X\), denoted as \(P(X)\), is given by:

\[ P(X = x) = \sum_{y} P(X = x, Y = y) \]

if \(Y\) is discrete, or

\[ P(X = x) = \int P(X = x, Y = y) \, dy \]

if \(Y\) is continuous.

Similarly, the marginal distribution of \(Y\) is:

\[ P(Y = y) = \sum_{x} P(X = x, Y = y) \]

if \(X\) is discrete, or

\[ P(Y = y) = \int P(X = x, Y = y) \, dx \]

if \(X\) is continuous.

Importance in Statistics

Marginal distributions are crucial in statistical analysis because they allow us to understand the behavior of individual variables within a multivariate distribution. They are used in various applications, such as:

• Bayesian statistics: Marginal distributions are used to compute posterior distributions by integrating over nuisance parameters.
• Regression analysis: Understanding the marginal distribution of a response variable can provide insights into its variability and central tendency.
• Machine learning: Marginal distributions are used in algorithms that require the estimation of probabilities for individual features.

Example

Consider a bivariate normal distribution with random variables \(X\) and \(Y\). The joint distribution is characterized by a mean vector and a covariance matrix. The marginal distributions of \(X\) and \(Y\) are both normal distributions, with means and variances derived from the joint distribution's parameters.

Related Concepts

• Conditional probability distribution
• Joint probability distribution
• Independence (probability theory)
• Covariance

Related Pages

• Probability distribution
• Random variable
• Multivariate normal distribution