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- [Stan Functions Reference](https://mc-stan.org/docs/functions-reference/index.html) - [*Overview*](https://mc-stan.org/docs/2_22/functions-reference/index.html#overview) - [Interfaces and Platforms](https://mc-stan.org/docs/2_22/functions-reference/index.html) - [Web Site](https://mc-stan.org/docs/2_22/functions-reference/index.html#web-site) - [GitHub Organization](https://mc-stan.org/docs/2_22/functions-reference/index.html#github-organization) - [Forums](https://mc-stan.org/docs/2_22/functions-reference/index.html#forums) - [Licensing](https://mc-stan.org/docs/2_22/functions-reference/index.html#licensing) - [Acknowledgements](https://mc-stan.org/docs/2_22/functions-reference/index.html#acknowledgements) - [*Built-In Functions*](https://mc-stan.org/docs/2_22/functions-reference/built-in-functions.html#built-in-functions) - [**1** Void Functions](https://mc-stan.org/docs/2_22/functions-reference/void-functions.html) - [**1\.1** Print Statement](https://mc-stan.org/docs/2_22/functions-reference/print-statement.html) - [**1\.2** Reject Statement](https://mc-stan.org/docs/2_22/functions-reference/reject-statement.html) - [**2** Integer-Valued Basic Functions](https://mc-stan.org/docs/2_22/functions-reference/integer-valued-basic-functions.html) - [**2\.1** Integer-Valued Arithmetic Operators](https://mc-stan.org/docs/2_22/functions-reference/int-arithmetic.html) - [**2\.1.1** Binary Infix Operators](https://mc-stan.org/docs/2_22/functions-reference/int-arithmetic.html#binary-infix-operators) - [**2\.1.2** Unary Prefix Operators](https://mc-stan.org/docs/2_22/functions-reference/int-arithmetic.html#unary-prefix-operators) - [**2\.2** Absolute Functions](https://mc-stan.org/docs/2_22/functions-reference/absolute-functions.html) - [**2\.3** Bound Functions](https://mc-stan.org/docs/2_22/functions-reference/bound-functions.html) - [**3** Real-Valued Basic Functions](https://mc-stan.org/docs/2_22/functions-reference/real-valued-basic-functions.html) - [**3\.1** Vectorization of Real-Valued Functions](https://mc-stan.org/docs/2_22/functions-reference/fun-vectorization.html) - [**3\.1.1** Unary Function Vectorization](https://mc-stan.org/docs/2_22/functions-reference/fun-vectorization.html#unary-function-vectorization) - [**3\.2** Mathematical Constants](https://mc-stan.org/docs/2_22/functions-reference/built-in-constants.html) - [**3\.3** Special Values](https://mc-stan.org/docs/2_22/functions-reference/special-values.html) - [**3\.4** Log Probability Function](https://mc-stan.org/docs/2_22/functions-reference/get-log-prob.html) - [**3\.5** Logical Functions](https://mc-stan.org/docs/2_22/functions-reference/logical-functions.html) - [**3\.5.1** Comparison Operators](https://mc-stan.org/docs/2_22/functions-reference/logical-functions.html#comparison-operators) - [**3\.5.2** Boolean Operators](https://mc-stan.org/docs/2_22/functions-reference/logical-functions.html#boolean-operators) - [**3\.5.3** Logical Functions](https://mc-stan.org/docs/2_22/functions-reference/logical-functions.html#logical-functions-1) - [**3\.6** Real-Valued Arithmetic Operators](https://mc-stan.org/docs/2_22/functions-reference/real-valued-arithmetic-operators.html) - [**3\.6.1** Binary Infix Operators](https://mc-stan.org/docs/2_22/functions-reference/real-valued-arithmetic-operators.html#binary-infix-operators-1) - [**3\.6.2** Unary Prefix Operators](https://mc-stan.org/docs/2_22/functions-reference/real-valued-arithmetic-operators.html#unary-prefix-operators-1) - [**3\.7** Step-like Functions](https://mc-stan.org/docs/2_22/functions-reference/step-functions.html) - [**3\.7.1** Absolute Value Functions](https://mc-stan.org/docs/2_22/functions-reference/step-functions.html#absolute-value-functions) - [**3\.7.2** Bounds Functions](https://mc-stan.org/docs/2_22/functions-reference/step-functions.html#bounds-functions) - [**3\.7.3** Arithmetic Functions](https://mc-stan.org/docs/2_22/functions-reference/step-functions.html#arithmetic-functions) - [**3\.7.4** Rounding Functions](https://mc-stan.org/docs/2_22/functions-reference/step-functions.html#rounding-functions) - [**3\.8** Power and Logarithm Functions](https://mc-stan.org/docs/2_22/functions-reference/power-and-logarithm-functions.html) - [**3\.9** Trigonometric Functions](https://mc-stan.org/docs/2_22/functions-reference/trigonometric-functions.html) - [**3\.10** Hyperbolic Trigonometric Functions](https://mc-stan.org/docs/2_22/functions-reference/hyperbolic-trigonometric-functions.html) - [**3\.11** Link Functions](https://mc-stan.org/docs/2_22/functions-reference/link-functions.html) - [**3\.12** Probability-Related Functions](https://mc-stan.org/docs/2_22/functions-reference/Phi-function.html) - [**3\.12.1** Normal Cumulative Distribution Functions](https://mc-stan.org/docs/2_22/functions-reference/Phi-function.html#normal-cumulative-distribution-functions) - [**3\.12.2** Other Probability-Related Functions](https://mc-stan.org/docs/2_22/functions-reference/Phi-function.html#other-probability-related-functions) - [**3\.13** Combinatorial Functions](https://mc-stan.org/docs/2_22/functions-reference/betafun.html) - [**3\.14** Composed Functions](https://mc-stan.org/docs/2_22/functions-reference/composed-functions.html) - [**4** Array Operations](https://mc-stan.org/docs/2_22/functions-reference/array-operations.html) - [**4\.1** Reductions](https://mc-stan.org/docs/2_22/functions-reference/array-reductions.html) - [**4\.1.1** Minimum and Maximum](https://mc-stan.org/docs/2_22/functions-reference/array-reductions.html#minimum-and-maximum) - [**4\.1.2** Sum, Product, and Log Sum of Exp](https://mc-stan.org/docs/2_22/functions-reference/array-reductions.html#sum-product-and-log-sum-of-exp) - [**4\.1.3** Sample Mean, Variance, and Standard Deviation](https://mc-stan.org/docs/2_22/functions-reference/array-reductions.html#sample-mean-variance-and-standard-deviation) - [**4\.1.4** Euclidean Distance and Squared Distance](https://mc-stan.org/docs/2_22/functions-reference/array-reductions.html#euclidean-distance-and-squared-distance) - [**4\.2** Array Size and Dimension Function](https://mc-stan.org/docs/2_22/functions-reference/array-size-and-dimension-function.html) - [**4\.3** Array Broadcasting](https://mc-stan.org/docs/2_22/functions-reference/array-broadcasting.html) - [**4\.4** Array Concatenation](https://mc-stan.org/docs/2_22/functions-reference/array-concatenation.html) - [**4\.5** Sorting functions](https://mc-stan.org/docs/2_22/functions-reference/sorting-functions.html) - [**5** Matrix Operations](https://mc-stan.org/docs/2_22/functions-reference/matrix-operations.html) - [**5\.1** Integer-Valued Matrix Size Functions](https://mc-stan.org/docs/2_22/functions-reference/integer-valued-matrix-size-functions.html) - [**5\.2** Matrix Arithmetic Operators](https://mc-stan.org/docs/2_22/functions-reference/matrix-arithmetic-operators.html) - [**5\.2.1** Negation Prefix Operators](https://mc-stan.org/docs/2_22/functions-reference/matrix-arithmetic-operators.html#negation-prefix-operators) - [**5\.2.2** Infix Matrix Operators](https://mc-stan.org/docs/2_22/functions-reference/matrix-arithmetic-operators.html#infix-matrix-operators) - [**5\.2.3** Broadcast Infix Operators](https://mc-stan.org/docs/2_22/functions-reference/matrix-arithmetic-operators.html#broadcast-infix-operators) - [**5\.2.4** Elementwise Arithmetic Operations](https://mc-stan.org/docs/2_22/functions-reference/matrix-arithmetic-operators.html#elementwise-arithmetic-operations) - [**5\.3** Transposition Operator](https://mc-stan.org/docs/2_22/functions-reference/transposition-operator.html) - [**5\.4** Elementwise Functions](https://mc-stan.org/docs/2_22/functions-reference/elementwise-functions.html) - [**5\.5** Dot Products and Specialized Products](https://mc-stan.org/docs/2_22/functions-reference/dot-products-and-specialized-products.html) - [**5\.5.1** Specialized Products](https://mc-stan.org/docs/2_22/functions-reference/dot-products-and-specialized-products.html#specialized-products) - [**5\.6** Reductions](https://mc-stan.org/docs/2_22/functions-reference/reductions.html) - [**5\.6.1** Log Sum of Exponents](https://mc-stan.org/docs/2_22/functions-reference/reductions.html#log-sum-of-exponents) - [**5\.6.2** Minimum and Maximum](https://mc-stan.org/docs/2_22/functions-reference/reductions.html#minimum-and-maximum-1) - [**5\.6.3** Sums and Products](https://mc-stan.org/docs/2_22/functions-reference/reductions.html#sums-and-products) - [**5\.6.4** Sample Moments](https://mc-stan.org/docs/2_22/functions-reference/reductions.html#sample-moments) - [**5\.7** Broadcast Functions](https://mc-stan.org/docs/2_22/functions-reference/matrix-broadcast.html) - [**5\.8** Diagonal Matrix Functions](https://mc-stan.org/docs/2_22/functions-reference/diagonal-matrix-functions.html) - [**5\.9** Slicing and Blocking Functions](https://mc-stan.org/docs/2_22/functions-reference/slicing-and-blocking-functions.html) - [**5\.9.1** Columns and Rows](https://mc-stan.org/docs/2_22/functions-reference/slicing-and-blocking-functions.html#columns-and-rows) - [**5\.9.2** Block Operations](https://mc-stan.org/docs/2_22/functions-reference/slicing-and-blocking-functions.html#block-operations) - [**5\.10** Matrix Concatenation](https://mc-stan.org/docs/2_22/functions-reference/matrix-concatenation.html) - [**5\.10.0.1** Horizontal concatenation](https://mc-stan.org/docs/2_22/functions-reference/matrix-concatenation.html#horizontal-concatenation) - [**5\.10.0.2** Vertical concatenation](https://mc-stan.org/docs/2_22/functions-reference/matrix-concatenation.html#vertical-concatenation) - [**5\.11** Special Matrix Functions](https://mc-stan.org/docs/2_22/functions-reference/softmax.html) - [**5\.11.1** Softmax](https://mc-stan.org/docs/2_22/functions-reference/softmax.html#softmax-1) - [**5\.11.2** Cumulative Sums](https://mc-stan.org/docs/2_22/functions-reference/softmax.html#cumulative-sums) - [**5\.12** Covariance Functions](https://mc-stan.org/docs/2_22/functions-reference/covariance.html) - [**5\.12.1** Exponentiated quadratic covariance function](https://mc-stan.org/docs/2_22/functions-reference/covariance.html#exponentiated-quadratic-covariance-function) - [**5\.13** Linear Algebra Functions and Solvers](https://mc-stan.org/docs/2_22/functions-reference/linear-algebra-functions-and-solvers.html) - [**5\.13.1** Matrix Division Operators and Functions](https://mc-stan.org/docs/2_22/functions-reference/linear-algebra-functions-and-solvers.html#matrix-division-operators-and-functions) - [**5\.13.2** Symmetric positive-definite matrix division functions](https://mc-stan.org/docs/2_22/functions-reference/linear-algebra-functions-and-solvers.html#symmetric-positive-definite-matrix-division-functions) - [**5\.13.3** Matrix Exponential](https://mc-stan.org/docs/2_22/functions-reference/linear-algebra-functions-and-solvers.html#matrix-exponential) - [**5\.13.4** Linear Algebra Functions](https://mc-stan.org/docs/2_22/functions-reference/linear-algebra-functions-and-solvers.html#linear-algebra-functions) - [**5\.14** Sort Functions](https://mc-stan.org/docs/2_22/functions-reference/sort-functions.html) - [**6** Sparse Matrix Operations](https://mc-stan.org/docs/2_22/functions-reference/sparse-matrices.html) - [**6\.1** Compressed Row Storage](https://mc-stan.org/docs/2_22/functions-reference/CSR.html) - [**6\.2** Conversion Functions](https://mc-stan.org/docs/2_22/functions-reference/conversion-functions.html) - [**6\.2.1** Dense to Sparse Conversion](https://mc-stan.org/docs/2_22/functions-reference/conversion-functions.html#dense-to-sparse-conversion) - [**6\.2.2** Sparse to Dense Conversion](https://mc-stan.org/docs/2_22/functions-reference/conversion-functions.html#sparse-to-dense-conversion) - [**6\.3** Sparse Matrix Arithmetic](https://mc-stan.org/docs/2_22/functions-reference/sparse-matrix-arithmetic.html) - [**6\.3.1** Sparse Matrix Multiplication](https://mc-stan.org/docs/2_22/functions-reference/sparse-matrix-arithmetic.html#sparse-matrix-multiplication) - [**7** Mixed Operations](https://mc-stan.org/docs/2_22/functions-reference/mixed-operations.html) - [**8** Compound Arithmetic and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-arithmetic-and-assignment.html) - [**8\.1** Compound Addition and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-addition-and-assignment.html) - [**8\.2** Compound Subtraction and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-subtraction-and-assignment.html) - [**8\.3** Compound Multiplication and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-multiplication-and-assignment.html) - [**8\.4** Compound Division and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-division-and-assignment.html) - [**8\.5** Compound Elementwise Multiplication and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-elementwise-multiplication-and-assignment.html) - [**8\.6** Compound Elementwise Division and Assignment](https://mc-stan.org/docs/2_22/functions-reference/compound-elementwise-division-and-assignment.html) - [**9** Higher-Order Functions](https://mc-stan.org/docs/2_22/functions-reference/higher-order-functions.html) - [**9\.1** Algebraic Equation Solver](https://mc-stan.org/docs/2_22/functions-reference/functions-algebraic-solver.html) - [**9\.1.1** Specifying an Algebraic Equation as a Function](https://mc-stan.org/docs/2_22/functions-reference/functions-algebraic-solver.html#specifying-an-algebraic-equation-as-a-function) - [**9\.1.2** Call to the Algebraic Solver](https://mc-stan.org/docs/2_22/functions-reference/functions-algebraic-solver.html#call-to-the-algebraic-solver) - [**9\.2** Ordinary Differential Equation (ODE) Solvers](https://mc-stan.org/docs/2_22/functions-reference/functions-ode-solver.html) - [**9\.2.1** Specifying an Ordinary Differential Equation as a Function](https://mc-stan.org/docs/2_22/functions-reference/functions-ode-solver.html#specifying-an-ordinary-differential-equation-as-a-function) - [**9\.2.2** Non-Stiff Solver](https://mc-stan.org/docs/2_22/functions-reference/functions-ode-solver.html#non-stiff-solver) - [**9\.2.3** Stiff Solver](https://mc-stan.org/docs/2_22/functions-reference/functions-ode-solver.html#stiff-solver) - [**9\.2.4** Arguments to the ODE solvers](https://mc-stan.org/docs/2_22/functions-reference/functions-ode-solver.html#arguments-to-the-ode-solvers) - [**9\.3** 1D Integrator](https://mc-stan.org/docs/2_22/functions-reference/functions-1d-integrator.html) - [**9\.3.1** Specifying an Integrand as a Function](https://mc-stan.org/docs/2_22/functions-reference/functions-1d-integrator.html#specifying-an-integrand-as-a-function) - [**9\.3.2** Call to the 1D Integrator](https://mc-stan.org/docs/2_22/functions-reference/functions-1d-integrator.html#call-to-the-1d-integrator) - [**9\.4** Higher-Order Map](https://mc-stan.org/docs/2_22/functions-reference/functions-map.html) - [**9\.4.1** Specifying the Mapped Function](https://mc-stan.org/docs/2_22/functions-reference/functions-map.html#specifying-the-mapped-function) - [**9\.4.2** Rectangular Map](https://mc-stan.org/docs/2_22/functions-reference/functions-map.html#rectangular-map) - [*Discrete Distributions*](https://mc-stan.org/docs/2_22/functions-reference/discrete-distributions.html#discrete-distributions) - [**10** Conventions for Probability Functions](https://mc-stan.org/docs/2_22/functions-reference/conventions-for-probability-functions.html) - [**10\.1** Suffix Marks Type of Function](https://mc-stan.org/docs/2_22/functions-reference/suffix-marks-type-of-function.html) - [**10\.2** Argument Order and the Vertical Bar](https://mc-stan.org/docs/2_22/functions-reference/argument-order-and-the-vertical-bar.html) - [**10\.3** Sampling Notation](https://mc-stan.org/docs/2_22/functions-reference/sampling-notation.html) - [**10\.4** Finite Inputs](https://mc-stan.org/docs/2_22/functions-reference/finite-inputs.html) - [**10\.5** Boundary Conditions](https://mc-stan.org/docs/2_22/functions-reference/boundary-conditions.html) - [**10\.6** Pseudorandom Number Generators](https://mc-stan.org/docs/2_22/functions-reference/distributions-prng.html) - [**10\.6.1** Restricted to Transformed Data and Generated Quantities](https://mc-stan.org/docs/2_22/functions-reference/distributions-prng.html#restricted-to-transformed-data-and-generated-quantities) - [**10\.6.2** Limited Vectorization](https://mc-stan.org/docs/2_22/functions-reference/distributions-prng.html#limited-vectorization) - [**10\.7** Cumulative Distribution Functions](https://mc-stan.org/docs/2_22/functions-reference/cumulative-distribution-functions.html) - [**10\.8** Vectorization](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html) - [**10\.8.1** Vectorized Function Signatures](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html#prob-vectorization) - [**10\.8.2** Evaluating Vectorized Log Probability Functions](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html#evaluating-vectorized-log-probability-functions) - [**10\.8.3** Evaluating Vectorized PRNG Functions](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html#prng-vectorization) - [**11** Binary Distributions](https://mc-stan.org/docs/2_22/functions-reference/binary-distributions.html) - [**11\.1** Bernoulli Distribution](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-distribution.html) - [**11\.1.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-distribution.html#probability-mass-function) - [**11\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-distribution.html#sampling-statement) - [**11\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-distribution.html#stan-functions) - [**11\.2** Bernoulli Distribution, Logit Parameterization](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-distribution.html) - [**11\.2.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-distribution.html#probability-mass-function-1) - [**11\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-distribution.html#sampling-statement-1) - [**11\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-distribution.html#stan-functions-1) - [**11\.3** Bernoulli-Logit Generalised Linear Model (Logistic Regression)](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-glm.html) - [**11\.3.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-glm.html#probability-mass-function-2) - [**11\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-glm.html#sampling-statement-2) - [**11\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/bernoulli-logit-glm.html#stan-functions-2) - [**12** Bounded Discrete Distributions](https://mc-stan.org/docs/2_22/functions-reference/bounded-discrete-distributions.html) - [**12\.1** Binomial Distribution](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html) - [**12\.1.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html#probability-mass-function-3) - [**12\.1.2** Log Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html#log-probability-mass-function) - [**12\.1.3** Gradient of Log Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html#gradient-of-log-probability-mass-function) - [**12\.1.4** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html#sampling-statement-3) - [**12\.1.5** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution.html#stan-functions-3) - [**12\.2** Binomial Distribution, Logit Parameterization](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html) - [**12\.2.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html#probability-mass-function-4) - [**12\.2.2** Log Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html#log-probability-mass-function-1) - [**12\.2.3** Gradient of Log Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html#gradient-of-log-probability-mass-function-1) - [**12\.2.4** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html#sampling-statement-4) - [**12\.2.5** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/binomial-distribution-logit-parameterization.html#stan-functions-4) - [**12\.3** Beta-Binomial Distribution](https://mc-stan.org/docs/2_22/functions-reference/beta-binomial-distribution.html) - [**12\.3.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/beta-binomial-distribution.html#probability-mass-function-5) - [**12\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/beta-binomial-distribution.html#sampling-statement-5) - [**12\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/beta-binomial-distribution.html#stan-functions-5) - [**12\.4** Hypergeometric Distribution](https://mc-stan.org/docs/2_22/functions-reference/hypergeometric-distribution.html) - [**12\.4.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/hypergeometric-distribution.html#probability-mass-function-6) - [**12\.4.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/hypergeometric-distribution.html#sampling-statement-6) - [**12\.4.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/hypergeometric-distribution.html#stan-functions-6) - [**12\.5** Categorical Distribution](https://mc-stan.org/docs/2_22/functions-reference/categorical-distribution.html) - [**12\.5.1** Probability Mass Functions](https://mc-stan.org/docs/2_22/functions-reference/categorical-distribution.html#probability-mass-functions) - [**12\.5.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/categorical-distribution.html#sampling-statement-7) - [**12\.5.3** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/categorical-distribution.html#sampling-statement-8) - [**12\.5.4** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/categorical-distribution.html#stan-functions-7) - [**12\.6** Ordered Logistic Distribution](https://mc-stan.org/docs/2_22/functions-reference/ordered-logistic-distribution.html) - [**12\.6.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/ordered-logistic-distribution.html#probability-mass-function-7) - [**12\.6.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/ordered-logistic-distribution.html#sampling-statement-9) - [**12\.6.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/ordered-logistic-distribution.html#stan-functions-8) - [**12\.7** Ordered Probit Distribution](https://mc-stan.org/docs/2_22/functions-reference/ordered-probit-distribution.html) - [**12\.7.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/ordered-probit-distribution.html#probability-mass-function-8) - [**12\.7.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/ordered-probit-distribution.html#sampling-statement-10) - [**12\.7.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/ordered-probit-distribution.html#stan-functions-9) - [**13** Unbounded Discrete Distributions](https://mc-stan.org/docs/2_22/functions-reference/unbounded-discrete-distributions.html) - [**13\.1** Negative Binomial Distribution](https://mc-stan.org/docs/2_22/functions-reference/negative-binomial-distribution.html) - [**13\.1.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/negative-binomial-distribution.html#probability-mass-function-9) - [**13\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/negative-binomial-distribution.html#sampling-statement-11) - [**13\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/negative-binomial-distribution.html#stan-functions-10) - [**13\.2** Negative Binomial Distribution (alternative parameterization)](https://mc-stan.org/docs/2_22/functions-reference/nbalt.html) - [**13\.2.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/nbalt.html#probability-mass-function-10) - [**13\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/nbalt.html#sampling-statement-12) - [**13\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/nbalt.html#stan-functions-11) - [**13\.3** Negative Binomial Distribution (log alternative parameterization)](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log.html) - [**13\.3.1** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log.html#sampling-statement-13) - [**13\.3.2** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log.html#stan-functions-12) - [**13\.4** Negative-Binomial-2-Log Generalised Linear Model (Negative Binomial Regression)](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log-glm.html) - [**13\.4.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log-glm.html#probability-mass-function-11) - [**13\.4.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log-glm.html#sampling-statement-14) - [**13\.4.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/neg-binom-2-log-glm.html#stan-functions-13) - [**13\.5** Poisson Distribution](https://mc-stan.org/docs/2_22/functions-reference/poisson.html) - [**13\.5.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/poisson.html#probability-mass-function-12) - [**13\.5.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/poisson.html#sampling-statement-15) - [**13\.5.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/poisson.html#stan-functions-14) - [**13\.6** Poisson Distribution, Log Parameterization](https://mc-stan.org/docs/2_22/functions-reference/poisson-distribution-log-parameterization.html) - [**13\.6.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/poisson-distribution-log-parameterization.html#probability-mass-function-13) - [**13\.6.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/poisson-distribution-log-parameterization.html#sampling-statement-16) - [**13\.6.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/poisson-distribution-log-parameterization.html#stan-functions-15) - [**13\.7** Poisson-Log Generalised Linear Model (Poisson Regression)](https://mc-stan.org/docs/2_22/functions-reference/poisson-log-glm.html) - [**13\.7.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/poisson-log-glm.html#probability-mass-function-14) - [**13\.7.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/poisson-log-glm.html#sampling-statement-17) - [**13\.7.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/poisson-log-glm.html#stan-functions-16) - [**14** Multivariate Discrete Distributions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-discrete-distributions.html) - [**14\.1** Multinomial Distribution](https://mc-stan.org/docs/2_22/functions-reference/multinomial-distribution.html) - [**14\.1.1** Probability Mass Function](https://mc-stan.org/docs/2_22/functions-reference/multinomial-distribution.html#probability-mass-function-15) - [**14\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multinomial-distribution.html#sampling-statement-18) - [**14\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multinomial-distribution.html#stan-functions-17) - [*Continuous Distributions*](https://mc-stan.org/docs/2_22/functions-reference/continuous-distributions.html#continuous-distributions) - [**15** Unbounded Continuous Distributions](https://mc-stan.org/docs/2_22/functions-reference/unbounded-continuous-distributions.html) - [**15\.1** Normal Distribution](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html) - [**15\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#probability-density-function) - [**15\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#sampling-statement-19) - [**15\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#stan-functions-18) - [**15\.1.4** Standard Normal Distribution](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#standard-normal-distribution) - [**15\.1.5** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#sampling-statement-20) - [**15\.1.6** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/normal-distribution.html#stan-functions-19) - [**15\.2** Normal-Id Generalised Linear Model (Linear Regression)](https://mc-stan.org/docs/2_22/functions-reference/normal-id-glm.html) - [**15\.2.1** Probability Distribution Function](https://mc-stan.org/docs/2_22/functions-reference/normal-id-glm.html#probability-distribution-function) - [**15\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/normal-id-glm.html#sampling-statement-21) - [**15\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/normal-id-glm.html#stan-functions-20) - [**15\.3** Exponentially Modified Normal Distribution](https://mc-stan.org/docs/2_22/functions-reference/exponentially-modified-normal-distribution.html) - [**15\.3.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/exponentially-modified-normal-distribution.html#probability-density-function-1) - [**15\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/exponentially-modified-normal-distribution.html#sampling-statement-22) - [**15\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/exponentially-modified-normal-distribution.html#stan-functions-21) - [**15\.4** Skew Normal Distribution](https://mc-stan.org/docs/2_22/functions-reference/skew-normal-distribution.html) - [**15\.4.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/skew-normal-distribution.html#probability-density-function-2) - [**15\.4.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/skew-normal-distribution.html#sampling-statement-23) - [**15\.4.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/skew-normal-distribution.html#stan-functions-22) - [**15\.5** Student-T Distribution](https://mc-stan.org/docs/2_22/functions-reference/student-t-distribution.html) - [**15\.5.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/student-t-distribution.html#probability-density-function-3) - [**15\.5.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/student-t-distribution.html#sampling-statement-24) - [**15\.5.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/student-t-distribution.html#stan-functions-23) - [**15\.6** Cauchy Distribution](https://mc-stan.org/docs/2_22/functions-reference/cauchy-distribution.html) - [**15\.6.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/cauchy-distribution.html#probability-density-function-4) - [**15\.6.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/cauchy-distribution.html#sampling-statement-25) - [**15\.6.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/cauchy-distribution.html#stan-functions-24) - [**15\.7** Double Exponential (Laplace) Distribution](https://mc-stan.org/docs/2_22/functions-reference/double-exponential-laplace-distribution.html) - [**15\.7.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/double-exponential-laplace-distribution.html#probability-density-function-5) - [**15\.7.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/double-exponential-laplace-distribution.html#sampling-statement-26) - [**15\.7.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/double-exponential-laplace-distribution.html#stan-functions-25) - [**15\.8** Logistic Distribution](https://mc-stan.org/docs/2_22/functions-reference/logistic-distribution.html) - [**15\.8.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/logistic-distribution.html#probability-density-function-6) - [**15\.8.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/logistic-distribution.html#sampling-statement-27) - [**15\.8.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/logistic-distribution.html#stan-functions-26) - [**15\.9** Gumbel Distribution](https://mc-stan.org/docs/2_22/functions-reference/gumbel-distribution.html) - [**15\.9.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/gumbel-distribution.html#probability-density-function-7) - [**15\.9.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/gumbel-distribution.html#sampling-statement-28) - [**15\.9.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/gumbel-distribution.html#stan-functions-27) - [**16** Positive Continuous Distributions](https://mc-stan.org/docs/2_22/functions-reference/positive-continuous-distributions.html) - [**16\.1** Lognormal Distribution](https://mc-stan.org/docs/2_22/functions-reference/lognormal.html) - [**16\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/lognormal.html#probability-density-function-8) - [**16\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/lognormal.html#sampling-statement-29) - [**16\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/lognormal.html#stan-functions-28) - [**16\.2** Chi-Square Distribution](https://mc-stan.org/docs/2_22/functions-reference/chi-square-distribution.html) - [**16\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/chi-square-distribution.html#probability-density-function-9) - [**16\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/chi-square-distribution.html#sampling-statement-30) - [**16\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/chi-square-distribution.html#stan-functions-29) - [**16\.3** Inverse Chi-Square Distribution](https://mc-stan.org/docs/2_22/functions-reference/inverse-chi-square-distribution.html) - [**16\.3.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/inverse-chi-square-distribution.html#probability-density-function-10) - [**16\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/inverse-chi-square-distribution.html#sampling-statement-31) - [**16\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/inverse-chi-square-distribution.html#stan-functions-30) - [**16\.4** Scaled Inverse Chi-Square Distribution](https://mc-stan.org/docs/2_22/functions-reference/scaled-inverse-chi-square-distribution.html) - [**16\.4.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/scaled-inverse-chi-square-distribution.html#probability-density-function-11) - [**16\.4.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/scaled-inverse-chi-square-distribution.html#sampling-statement-32) - [**16\.4.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/scaled-inverse-chi-square-distribution.html#stan-functions-31) - [**16\.5** Exponential Distribution](https://mc-stan.org/docs/2_22/functions-reference/exponential-distribution.html) - [**16\.5.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/exponential-distribution.html#probability-density-function-12) - [**16\.5.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/exponential-distribution.html#sampling-statement-33) - [**16\.5.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/exponential-distribution.html#stan-functions-32) - [**16\.6** Gamma Distribution](https://mc-stan.org/docs/2_22/functions-reference/gamma-distribution.html) - [**16\.6.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/gamma-distribution.html#probability-density-function-13) - [**16\.6.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/gamma-distribution.html#sampling-statement-34) - [**16\.6.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/gamma-distribution.html#stan-functions-33) - [**16\.7** Inverse Gamma Distribution](https://mc-stan.org/docs/2_22/functions-reference/inverse-gamma-distribution.html) - [**16\.7.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/inverse-gamma-distribution.html#probability-density-function-14) - [**16\.7.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/inverse-gamma-distribution.html#sampling-statement-35) - [**16\.7.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/inverse-gamma-distribution.html#stan-functions-34) - [**16\.8** Weibull Distribution](https://mc-stan.org/docs/2_22/functions-reference/weibull-distribution.html) - [**16\.8.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/weibull-distribution.html#probability-density-function-15) - [**16\.8.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/weibull-distribution.html#sampling-statement-36) - [**16\.8.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/weibull-distribution.html#stan-functions-35) - [**16\.9** Frechet Distribution](https://mc-stan.org/docs/2_22/functions-reference/frechet-distribution.html) - [**16\.9.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/frechet-distribution.html#probability-density-function-16) - [**16\.9.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/frechet-distribution.html#sampling-statement-37) - [**16\.9.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/frechet-distribution.html#stan-functions-36) - [**16\.10** Rayleigh Distribution](https://mc-stan.org/docs/2_22/functions-reference/rayleigh-distribution.html) - [**16\.10.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/rayleigh-distribution.html#probability-density-function-17) - [**16\.10.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/rayleigh-distribution.html#sampling-statement-38) - [**16\.10.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/rayleigh-distribution.html#stan-functions-37) - [**17** Positive Lower-Bounded Distributions](https://mc-stan.org/docs/2_22/functions-reference/positive-lower-bounded-distributions.html) - [**17\.1** Pareto Distribution](https://mc-stan.org/docs/2_22/functions-reference/pareto-distribution.html) - [**17\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/pareto-distribution.html#probability-density-function-18) - [**17\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/pareto-distribution.html#sampling-statement-39) - [**17\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/pareto-distribution.html#stan-functions-38) - [**17\.2** Pareto Type 2 Distribution](https://mc-stan.org/docs/2_22/functions-reference/pareto-type-2-distribution.html) - [**17\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/pareto-type-2-distribution.html#probability-density-function-19) - [**17\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/pareto-type-2-distribution.html#sampling-statement-40) - [**17\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/pareto-type-2-distribution.html#stan-functions-39) - [**17\.3** Wiener First Passage Time Distribution](https://mc-stan.org/docs/2_22/functions-reference/wiener-first-passage-time-distribution.html) - [**17\.3.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/wiener-first-passage-time-distribution.html#probability-density-function-20) - [**17\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/wiener-first-passage-time-distribution.html#sampling-statement-41) - [**17\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/wiener-first-passage-time-distribution.html#stan-functions-40) - [**17\.3.4** Boundaries](https://mc-stan.org/docs/2_22/functions-reference/wiener-first-passage-time-distribution.html#boundaries) - [**18** Continuous Distributions on \[0, 1\]](https://mc-stan.org/docs/2_22/functions-reference/continuous-distributions-on-0-1.html) - [**18\.1** Beta Distribution](https://mc-stan.org/docs/2_22/functions-reference/beta-distribution.html) - [**18\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/beta-distribution.html#probability-density-function-21) - [**18\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/beta-distribution.html#sampling-statement-42) - [**18\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/beta-distribution.html#stan-functions-41) - [**18\.2** Beta Proportion Distribution](https://mc-stan.org/docs/2_22/functions-reference/beta-proportion-distribution.html) - [**18\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/beta-proportion-distribution.html#probability-density-function-22) - [**18\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/beta-proportion-distribution.html#sampling-statement-43) - [**18\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/beta-proportion-distribution.html#stan-functions-42) - [**19** Circular Distributions](https://mc-stan.org/docs/2_22/functions-reference/circular-distributions.html) - [**19\.1** Von Mises Distribution](https://mc-stan.org/docs/2_22/functions-reference/von-mises-distribution.html) - [**19\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/von-mises-distribution.html#probability-density-function-23) - [**19\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/von-mises-distribution.html#sampling-statement-44) - [**19\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/von-mises-distribution.html#stan-functions-43) - [**19\.1.4** Numerical Stability](https://mc-stan.org/docs/2_22/functions-reference/von-mises-distribution.html#numerical-stability) - [**20** Bounded Continuous Distributions](https://mc-stan.org/docs/2_22/functions-reference/bounded-continuous-distributions.html) - [**20\.1** Uniform Distribution](https://mc-stan.org/docs/2_22/functions-reference/uniform-distribution.html) - [**20\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/uniform-distribution.html#probability-density-function-24) - [**20\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/uniform-distribution.html#sampling-statement-45) - [**20\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/uniform-distribution.html#stan-functions-44) - [**21** Distributions over Unbounded Vectors](https://mc-stan.org/docs/2_22/functions-reference/distributions-over-unbounded-vectors.html) - [**21\.1** Multivariate Normal Distribution](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution.html) - [**21\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution.html#probability-density-function-25) - [**21\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution.html#sampling-statement-46) - [**21\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution.html#stan-functions-45) - [**21\.2** Multivariate Normal Distribution, Precision Parameterization](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution-precision-parameterization.html) - [**21\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution-precision-parameterization.html#probability-density-function-26) - [**21\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution-precision-parameterization.html#sampling-statement-47) - [**21\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-normal-distribution-precision-parameterization.html#stan-functions-46) - [**21\.3** Multivariate Normal Distribution, Cholesky Parameterization](https://mc-stan.org/docs/2_22/functions-reference/multi-normal-cholesky-fun.html) - [**21\.3.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multi-normal-cholesky-fun.html#probability-density-function-27) - [**21\.3.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multi-normal-cholesky-fun.html#sampling-statement-48) - [**21\.3.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multi-normal-cholesky-fun.html#stan-functions-47) - [**21\.4** Multivariate Gaussian Process Distribution](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution.html) - [**21\.4.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution.html#probability-density-function-28) - [**21\.4.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution.html#sampling-statement-49) - [**21\.4.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution.html#stan-functions-48) - [**21\.5** Multivariate Gaussian Process Distribution, Cholesky parameterization](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution-cholesky-parameterization.html) - [**21\.5.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution-cholesky-parameterization.html#probability-density-function-29) - [**21\.5.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution-cholesky-parameterization.html#sampling-statement-50) - [**21\.5.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-gaussian-process-distribution-cholesky-parameterization.html#stan-functions-49) - [**21\.6** Multivariate Student-T Distribution](https://mc-stan.org/docs/2_22/functions-reference/multivariate-student-t-distribution.html) - [**21\.6.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/multivariate-student-t-distribution.html#probability-density-function-30) - [**21\.6.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/multivariate-student-t-distribution.html#sampling-statement-51) - [**21\.6.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/multivariate-student-t-distribution.html#stan-functions-50) - [**21\.7** Gaussian Dynamic Linear Models](https://mc-stan.org/docs/2_22/functions-reference/gaussian-dynamic-linear-models.html) - [**21\.7.1** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/gaussian-dynamic-linear-models.html#sampling-statement-52) - [**21\.7.2** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/gaussian-dynamic-linear-models.html#stan-functions-51) - [**22** Simplex Distributions](https://mc-stan.org/docs/2_22/functions-reference/simplex-distributions.html) - [**22\.1** Dirichlet Distribution](https://mc-stan.org/docs/2_22/functions-reference/dirichlet-distribution.html) - [**22\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/dirichlet-distribution.html#probability-density-function-31) - [**22\.1.2** Meaning of Dirichlet Parameters](https://mc-stan.org/docs/2_22/functions-reference/dirichlet-distribution.html#meaning-of-dirichlet-parameters) - [**22\.1.3** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/dirichlet-distribution.html#sampling-statement-53) - [**22\.1.4** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/dirichlet-distribution.html#stan-functions-52) - [**23** Correlation Matrix Distributions](https://mc-stan.org/docs/2_22/functions-reference/correlation-matrix-distributions.html) - [**23\.1** LKJ Correlation Distribution](https://mc-stan.org/docs/2_22/functions-reference/lkj-correlation.html) - [**23\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/lkj-correlation.html#probability-density-function-32) - [**23\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/lkj-correlation.html#sampling-statement-54) - [**23\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/lkj-correlation.html#stan-functions-53) - [**23\.2** Cholesky LKJ Correlation Distribution](https://mc-stan.org/docs/2_22/functions-reference/cholesky-lkj-correlation-distribution.html) - [**23\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/cholesky-lkj-correlation-distribution.html#probability-density-function-33) - [**23\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/cholesky-lkj-correlation-distribution.html#sampling-statement-55) - [**23\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/cholesky-lkj-correlation-distribution.html#stan-functions-54) - [**24** Covariance Matrix Distributions](https://mc-stan.org/docs/2_22/functions-reference/covariance-matrix-distributions.html) - [**24\.1** Wishart Distribution](https://mc-stan.org/docs/2_22/functions-reference/wishart-distribution.html) - [**24\.1.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/wishart-distribution.html#probability-density-function-34) - [**24\.1.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/wishart-distribution.html#sampling-statement-56) - [**24\.1.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/wishart-distribution.html#stan-functions-55) - [**24\.2** Inverse Wishart Distribution](https://mc-stan.org/docs/2_22/functions-reference/inverse-wishart-distribution.html) - [**24\.2.1** Probability Density Function](https://mc-stan.org/docs/2_22/functions-reference/inverse-wishart-distribution.html#probability-density-function-35) - [**24\.2.2** Sampling Statement](https://mc-stan.org/docs/2_22/functions-reference/inverse-wishart-distribution.html#sampling-statement-57) - [**24\.2.3** Stan Functions](https://mc-stan.org/docs/2_22/functions-reference/inverse-wishart-distribution.html#stan-functions-56) - [*Appendix*](https://mc-stan.org/docs/2_22/functions-reference/appendix.html#appendix) - [**25** Mathematical Functions](https://mc-stan.org/docs/2_22/functions-reference/math-functions.html) - [**25\.1** Beta](https://mc-stan.org/docs/2_22/functions-reference/beta-appendix.html) - [**25\.2** Incomplete Beta](https://mc-stan.org/docs/2_22/functions-reference/inc-beta-appendix.html) - [**25\.3** Gamma](https://mc-stan.org/docs/2_22/functions-reference/gamma-appendix.html) - [**25\.4** Digamma](https://mc-stan.org/docs/2_22/functions-reference/digamma-appendix.html) - [*References*](https://mc-stan.org/docs/2_22/functions-reference/references.html#references) # [Stan Functions Reference](https://mc-stan.org/docs/2_22/functions-reference/) This is an old version, [view current version](https://mc-stan.org/docs/functions-reference/continuous_distributions_on_0_1.html#beta-distribution). ## 18\.1 Beta Distribution ### 18\.1.1 Probability Density Function If α∈R\+ α ∈ R \+ and β∈R\+ β ∈ R \+, then for θ∈(0,1) θ ∈ ( 0 , 1 ), Beta(θ\|α,β)\=1B(α,β)θα−1(1−θ)β−1, Beta ( θ \| α , β ) \= 1 B ( α , β ) θ α − 1 ( 1 − θ ) β − 1 , where the beta function B() B ( ) is as defined in section [combinatorial functions](https://mc-stan.org/docs/2_22/functions-reference/betafun.html#betafun). ***Warning:*** If θ\=0 θ \= 0 or θ\=1 θ \= 1, then the probability is 0 and the log probability is −∞ − ∞. Similarly, the distribution requires strictly positive parameters, α,β\>0 α , β \> 0. ### 18\.1.2 Sampling Statement `theta ~` **`beta`**`(alpha, beta)` Increment target log probability density with `beta_lpdf(theta | alpha, beta)` dropping constant additive terms. ### 18\.1.3 Stan Functions `real` **`beta_lpdf`**`(reals theta | reals alpha, reals beta)` The log of the beta density of `theta` in \[0,1\] \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_cdf`**`(reals theta, reals alpha, reals beta)` The beta cumulative distribution function of `theta` in \[0,1\] \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_lcdf`**`(reals theta | reals alpha, reals beta)` The log of the beta cumulative distribution function of `theta` in \[0,1\] \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_lccdf`**`(reals theta | reals alpha, reals beta)` The log of the beta complementary cumulative distribution function of `theta` in \[0,1\] \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `R` **`beta_rng`**`(reals alpha, reals beta)` Generate a beta variate with positive prior successes (plus one) alpha and prior failures (plus one) beta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section [vectorized PRNG functions](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html#prng-vectorization).

### Probability Density Function If α ∈ R \+ and β ∈ R \+, then for θ ∈ ( 0 , 1 ), Beta ( θ \| α , β ) \= 1 B ( α , β ) θ α − 1 ( 1 − θ ) β − 1 , where the beta function B ( ) is as defined in section [combinatorial functions](https://mc-stan.org/docs/2_22/functions-reference/betafun.html#betafun). ***Warning:*** If θ \= 0 or θ \= 1, then the probability is 0 and the log probability is − ∞. Similarly, the distribution requires strictly positive parameters, α , β \> 0. ### Sampling Statement `theta ~` **`beta`**`(alpha, beta)` Increment target log probability density with `beta_lpdf(theta | alpha, beta)` dropping constant additive terms. ### Stan Functions `real` **`beta_lpdf`**`(reals theta | reals alpha, reals beta)` The log of the beta density of `theta` in \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_cdf`**`(reals theta, reals alpha, reals beta)` The beta cumulative distribution function of `theta` in \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_lcdf`**`(reals theta | reals alpha, reals beta)` The log of the beta cumulative distribution function of `theta` in \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `real` **`beta_lccdf`**`(reals theta | reals alpha, reals beta)` The log of the beta complementary cumulative distribution function of `theta` in \[ 0 , 1 \] given positive prior successes (plus one) alpha and prior failures (plus one) beta `R` **`beta_rng`**`(reals alpha, reals beta)` Generate a beta variate with positive prior successes (plus one) alpha and prior failures (plus one) beta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section [vectorized PRNG functions](https://mc-stan.org/docs/2_22/functions-reference/vectorization.html#prng-vectorization).