NormalDistribution

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

NormalDistribution[mu, sigma]

represents a normal distribution; NormalDistribution[] is the standard normal. Unlike a fitted model it prints its parameters in full, because they are what the user specified rather than an implementation detail.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

A specified distribution prints its parameters in full

In[1]:= NormalDistribution[2, 3]
Out[1]= NormalDistribution[2, 3]

The standard-normal density at 0 is 1/Sqrt[2 Pi]

In[2]:= PDF[NormalDistribution[0, 1], 0]
Out[2]= 0.398942

And away from the mean

In[3]:= PDF[NormalDistribution[0, 1], 1]
Out[3]= 0.241971

NormalDistribution[] is the standard normal

In[4]:= PDF[NormalDistribution[], 0]
Out[4]= 0.398942

Implementation notes

Definition. NormalDistribution[mu, sigma] represents a normal (Gaussian) distribution with mean mu and standard deviation sigma; NormalDistribution[] is the standard normal (mu = 0, sigma = 1). It is an inert, Protected distribution head — no builtin of its own — consumed by PDF and RandomVariate. Unlike a fitted LearnedDistribution, it prints its parameters in full, because they are what the user specified. Its docstring is in dist.c.

Representation. The head stays symbolic until a distribution builtin reads it. ml_read_dist (src/ml/dist.c) recognises NormalDistribution: with no arguments it is the standard normal; with two it reads mu and sigma as real scalars into a small MlDist record (kind ML_D_NORMAL). A non-positive sigma makes the read decline rather than produce NaNs. PDF then evaluates exp(-z^2/2)/(sigma sqrt(2 pi)) with z = (x - mu)/sigma, and RandomVariate draws from the same random stream as RandomReal.

Usage & limits. A specification, not a value — it carries no density on its own; pass it to PDF[dist, x] (which threads over a list of points) or RandomVariate[dist, n]. Parameters and the evaluation point must be numeric for PDF/RandomVariate; a symbolic point leaves PDF unevaluated.

Attributes: Protected.

References

• Source: src/ml/dist.c
• Specification index: Mathilda_spec.md
• Tests: tests/test_ml_classify.c
• Tests: tests/test_ml_dist.c

Notes & additional examples

Notes

NormalDistribution[mu, sigma] is a specification, not a value: it stays symbolic and prints its parameters in full (unlike a fitted LearnedDistribution, which prints elided). Pass it to PDF[dist, x] for the density exp(-z^2/2)/(sigma Sqrt[2 Pi]) with z = (x - mu)/sigma, or to RandomVariate[dist, n] for samples (from the same stream as RandomReal, so SeedRandom makes them reproducible). NormalDistribution[] is the standard normal. A non-positive sigma, or a non-numeric evaluation point, leaves the consuming call unevaluated rather than producing NaNs.