Random variable

As explained in Bayesian Optimization in Action

A random variable is a variable whose value is determined by an uncertain outcome or probabilistic process.

A random variable represents a quantity whose value comes from an uncertain outcome or probabilistic process. In multivariate modeling, several random variables are collected into a vector so their individual behavior and correlations can be modeled together. section 2.2.1

Within a multivariate Gaussian, each random variable has:

• An expected value given by the corresponding element of the mean vector. section 2.2.1
• A variance given by the corresponding diagonal entry of the covariance matrix. section 2.2.1
• Covariances with other variables represented by off-diagonal matrix entries. section 2.2.1

Everyday quantities

Height, IQ, income, and birth weight can be modeled as random variables; a housing-price example shows several such variables jointly through samples and confidence intervals. section 2.2.1

Figure 2.4 An MVN distribution groups multiple normally distributed random variables together. While the mean vector of the MVN concatenates the mean values, the covariance matrix models the correlations between the individual variables.

Visualizes how multiple normally distributed random variables form a multivariate Gaussian with a mean vector and covariance matrix. section 2.2.1

How Bayesian Optimization in Action covers it

How deep it goes

Defines variables within a Gaussian vector and traces their means, variances, and covariances through matrix entries.

Examples

Illustrates random variables with height, IQ, income, birth weight, and a housing-price multivariate example.

What you can do after

Evaluate how Gaussian models assign expected values, variances, and covariances to individual uncertain quantities.

How it connects

Connects random variables to normal distributions, multivariate Gaussian distributions, mean vectors, and covariance matrices.

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Also explained in

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  Math and Architectures of Deep LearningA quantity whose possible values are associated with probabilities, used to model uncertain measurable attributes or event outcomes.
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  Statistics Every Programmer NeedsA random variable is a function that assigns numerical values to outcomes of a random experiment.
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  Causal AIA random variable is a variable whose possible values and uncertainty are described by a probability distribution.
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  Causal Inference for Data ScienceA random variable is a variable whose value is determined by the outcome of a random experiment.

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Related concepts

• probability distributionused by · A probability distribution is a mathematical specification assigning probabilities to the possible values or outcomes of a random variable, with total probability one.
• interventionconfigured by · An intervention is a deliberate, externally imposed change or control applied to a variable or system to alter or assess its behavior or outcomes.
• distanceincludes · Distance is a measure of separation between points or entities in a defined space or context.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.