Distributions

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

published · difficulty 2/5 · 15 min read

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Intuition

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PredictName the object in plain language, then predict what should change.Leave with one reusable mental picture before notation appears.

The raw outcome might be $HH$ , $HT$ , $TH$ , or $TT$ , but your model only records $X=$ number of heads. So what is the probability of $X=1$ ?

A random variable is the measurement rule that turns each raw outcome into a value. A distribution is the probability measure induced on those values after the rule is applied.

The distinction is small but important. The world may have detailed outcomes: full coin-flip sequences, images, documents, users, or physical states. A random variable turns each outcome into a value we care about. The distribution tells us how probability lands on those values. "Measurement" is only a metaphor: $X$ does not have to be a physical instrument, and the distribution is not the rule itself. It is the probability law created by applying the rule to uncertain outcomes.

Many different outcomes can map to the same value. If $X$ is the number of heads in two coin flips, both $HT$ and $TH$ become $X=1$ . The probability of $X=1$ is the combined probability of all outcomes that map there.

That is the bridge to modeling. Once you know the distribution of a random variable, you can sample from it, compute expectations, and score observations. Maximum likelihood will later ask which parameter setting makes the observed values most probable under a chosen distribution family.

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Math

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Equation 1Inspect equation objectX:(\Omega,\mathcal F)\to(S,\mathcal S)Equation 2Inspect equation objectP_X(A)=P(X\in A)=P(\{\omega\in\Omega:X(\omega)\in A\}).Equation 3Inspect equation objectp_X(x)=P(X=x).

Let $(\Omega,\mathcal F,P)$ be a probability space. Let $(S,\mathcal S)$ be the measurable space of values, and let

X:(\Omega,\mathcal F)\to(S,\mathcal S)

be a measurable random variable. The distribution, or law, of $X$ is the probability measure $P_X$ on values:

P_X(A)=P(X\in A)=P(\{\omega\in\Omega:X(\omega)\in A\}).

Here $A\in\mathcal S$ is a measurable set of values. Equivalently, $P_X(A)=P(X^{-1}(A))$ .

This is called the pushforward of $P$ through $X$ . It says: start with probability on raw outcomes, apply the measurement $X$ , and add up the probability that lands inside each value-set $A$ .

For a discrete random variable, the probability mass function is

p_X(x)=P(X=x).

If the raw sample space is finite, this is just the mass of every outcome that maps to $x$ :

p_X(x)=\sum_{\omega:X(\omega)=x}P(\omega).

This is the central mechanism: a value collects probability from every raw outcome that maps to it. Grouping changes the representation of probability, not the total mass, so $\sum_x p_X(x)=1$ .

When $X$ is real-valued, the cumulative distribution function is

F_X(t)=P(X\le t).

It accumulates the mass up to a threshold. For many real-valued distributions used in machine learning, $P_X$ has a density $f_X$ and interval probabilities are computed by integration:

P(a\le X\le b)=\int_a^b f_X(x)\,dx.

A density value is not itself a probability; only area under the density over a region is probability. Density values can be greater than $1$ , because they depend on the units of $x$ . Some distributions are mixed or have no ordinary density, so PMFs and densities are important cases, not the whole definition of distribution.

The demo uses two independent Bernoulli flips with head probability $p$ and random variable $X=$ number of heads. Then $X$ has a binomial distribution:

P(X=k)=\binom{2}{k}p^k(1-p)^{2-k},\qquad k\in\{0,1,2\}.

Its expectation and variance are

\mathbb E[X]=2p,\qquad \operatorname{Var}(X)=2p(1-p).

In machine learning, a parametric distribution family writes this mass or density as $p_\theta(x)$ . If observed values $x^{(1)},\dots,x^{(n)}$ are treated as independent draws from the distribution, the log likelihood is

\ell(\theta)=\sum_{i=1}^n \log p_\theta(x^{(i)}).

For discrete data, $p_\theta(x)$ is a probability mass. For continuous data, $p_\theta(x)$ is a density value, so likelihood scores density at the observations; it is not the probability of observing those exact points.

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Code

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Code witness 1import math import numpy as np p_head = 0.65 # Raw outcomes for two independent flips. outcom...python

import math
import numpy as np

p_head = 0.65

# Raw outcomes for two independent flips.
outcomes = {
    "HH": p_head * p_head,
    "HT": p_head * (1 - p_head),
    "TH": (1 - p_head) * p_head,
    "TT": (1 - p_head) * (1 - p_head),
}

def number_of_heads(outcome):
    return outcome.count("H")

# Push probability mass through X(outcome) = number of heads.
pmf = {0: 0.0, 1: 0.0, 2: 0.0}
for outcome, prob in outcomes.items():
    pmf[number_of_heads(outcome)] += prob

# Grouping changes where the mass lives, not the total amount of mass.
assert abs(sum(outcomes.values()) - 1.0) < 1e-12
assert abs(sum(pmf.values()) - 1.0) < 1e-12

mean = sum(x * prob for x, prob in pmf.items())
variance = sum((x - mean) ** 2 * prob for x, prob in pmf.items())

print("PMF:", {x: round(prob, 3) for x, prob in pmf.items()})
print("E[X]:", round(mean, 3))
print("Var(X):", round(variance, 3))

# Likelihood scores observed values under this distribution.
observed = np.array([2, 1, 1, 0, 2])
observed_masses = [pmf[int(x)] for x in observed]
log_likelihood = (
    -math.inf
    if any(prob == 0 for prob in observed_masses)
    else sum(math.log(prob) for prob in observed_masses)
)
print("observed values:", observed.tolist())
print("log likelihood:", log_likelihood if math.isinf(log_likelihood) else round(log_likelihood, 3))

The code does not start with the binomial formula. It first assigns probability to raw outcomes, applies the random variable, and adds the mass that lands on the same value. The closed-form binomial PMF is the compact result of that aggregation.

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Interactive Demo

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ManipulateChange one control and predict the visible response before reveal.Leave with the observed invariant or a repaired model.

Live Concept Demo

Explore Distributions

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 2/5undergraduatecode-aligned

Demo inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity

02ObserveDemo state pending

03GroundName the equation, invariant, or control that explains it.

04CarryNext: Maximum Likelihood

Choose what to inspect in Distributions. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

Move the head-probability slider. The map from outcomes to $X$ stays fixed, but the probability mass on outcomes changes, so the distribution of $X$ changes.

Watch two things at once: the left side shows raw outcome probabilities, while the right side shows the aggregated probability mass function. The readout connects the distribution to expectation, variance, and a small i.i.d. observed-data log likelihood. At the endpoints, some observed values become impossible, so the log likelihood falls to negative infinity.

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Object

4/4 sections ready

Concept: Distributions

What is the smallest example that makes Distributions click without losing the math?

BeforeRandom VariablesNow4/4 sections readyTryManipulate one control and predict the visible change.NextMaximum Likelihood

Object contextProbability

StateProbability

2 sources attachedChecking local snapshot

concept:probability/distributions

Math3 equation objectsX:(\Omega,\mathcal F)\to(S,\mathcal S)CodeDistributions code witness 1assert abs(sum(outcomes.values()) - 1.0) < 1e-12 DemoDistributions predictionManipulate one control and predict the visible change.

ConceptLearner lens

Distributions

What is the smallest example that makes Distributions click without losing the math?

Mode questionCan I say the mechanism back in one sentence before I reveal anything?

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Take this move

Study modes

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01PredictCommit before reveal.02ExplainName the invariant.03Ground2 source links nearby.04ExperimentChange one knob.05CarrySave the next question.

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4/4 sections ready

Carry inRandom Variables

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Work hereDistributions

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

Carry outMaximum Likelihood

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After The First Pass

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The lower panels are one second act: keep the object fixed, inspect it visually, check source boundaries, practice transfer, then attach the research question.

ConceptDistributionsProbability

01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary2 source links attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

Mechanism Storyboard

See the idea move before the page explains it

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

Demo notes open01 / Intuition

Editorial probability illustration of probability mass pushed through a random variable into PMF bars and density curves.

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Distributions should make visible.

Visual Inquiry

Make the image answer a mathematical question

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

4/4 stages readyDemo notes connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Distributions easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

Object - ConceptDistributionsQuestion

What is the smallest example that makes Distributions click without losing the math?

concept:probability/distributions

Boundary

sources: deisenroth-2020-mml, goodfellow-2016-deep-learning

Check

Open the closest source note before trusting the local explanation.

Evidence

2 selected-object sources shown first; 2 references total.

Next move

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selected object source · book · 2020Mathematics for Machine LearningDeisenroth, Faisal, and Ong

Located CF editorial boundary

Grounds probability distributions, densities, expectations, and the notation used in ML models.

Used here as

Mathematics for Machine Learning grounds random variables, distributions, densities, and expectations; Goodfellow et al. use the same probability vocabulary as the base language for proba...

Caveat

This checks the probability-law framing and common PMF/density vocabulary, not measure-theoretic edge cases or every distribution family used in ML.

Open source

selected object source · book · 2016Deep LearningGoodfellow, Bengio, and Courville

Located CF editorial boundary

Grounds the probability and information-theory vocabulary reused by generative modeling pages.

Used here as

Mathematics for Machine Learning grounds random variables, distributions, densities, and expectations; Goodfellow et al. use the same probability vocabulary as the base language for proba...

Caveat

This checks the probability-law framing and common PMF/density vocabulary, not measure-theoretic edge cases or every distribution family used in ML.

Open source

Claim Review

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

Object - ConceptDistributionsQuestion

What is the smallest example that makes Distributions click without losing the math?

concept:probability/distributions

Boundary

sources: deisenroth-2020-mml, goodfellow-2016-deep-learning

Check

Treat every claim as provisional until source support and a local witness agree.

Evidence

1 structured claim check on this concept.

Next move

Run the prediction or practice transfer before asking for a grounded review.

1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 2 references and 3 local witnesses are available for inspection.

A distribution is the probability law induced by a random variable: it pushes probability from raw outcomes onto measurable value sets, with PMFs and densities as common representations.

Registered CF editorial boundary

Mathematics for Machine LearningGrounds probability distributions, densities, expectations, and the notation used in ML models.Deep LearningGrounds the probability and information-theory vocabulary reused by generative modeling pages.

Used here as

Mathematics for Machine Learning grounds random variables, distributions, densities, and expectations; Goodfellow et al. use the same probability vocabulary as the base language for probabilistic models and...

Local witness

source-span-deisenroth-2020-mml source-span-goodfellow-2016-deep-learning math-object-1 math-object-2 code-witness-1 interactive-demo

Equation 1

X:(\Omega,\mathcal F)\to(S,\mathcal S)

Equation 2

P_X(A)=P(X\in A)=P(\{\omega\in\Omega:X(\omega)\in A\}).

Caveat

This checks the probability-law framing and common PMF/density vocabulary, not measure-theoretic edge cases or every distribution family used in ML.

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Checked MML chapter 6.1-6.2 and Goodfellow chapter 3.3: MML defines a random variable as a map from outcomes to a target space, gives PX(S)=P(X in S)=P(X^-1(S)), and calls PX, or P composed with X^-1, the law/distribution of X. MML and Goodfellow both distinguish discrete PMFs from continuous densities/PDFs, with density probabilities obtained by integration over value sets.

Reviewer: codex+oracle; reviewed 2026-05-06

Practice · Distributions

Try the idea in your own words

A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

Concept · Current object

Distributions

Source boundary: sources: deisenroth-2020-mml, goodfellow-2016-deep-learning

Object context and links

Probability

concept:probability/distributions

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Explain the mechanism

For Distributions: What is the smallest example that makes Distributions click without losing the math? Explain your answer, including what changes, why, and which assumption matters.

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Grounded object roomClose

Selected object routeAsk from this object; carry one invariant back.sources: deisenroth-2020-mml, goodfellow-2016-deep-learning

ObjectConceptDistributions
PredictBefore revealDistributions prediction
WitnessCompare codeDistributions code witness 1
RoomAsk groundedChecking local snapshot

ConceptDistributionsProbability

Code witness comparisonDistributions code witness 1assert abs(sum(outcomes.values()) - 1.0) < 1e-12 Prediction before revealDistributions predictionManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Distributions click without losing the math?Checking local snapshot

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conceptProbability

Distributions

Anchored question

What is the smallest example that makes Distributions click without losing the math?

Source boundaryInspect source ids: deisenroth-2020-mml, goodfellow-2016-deep-learningStable content-object key attached

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Learner evidence requestAsk what would make "Distributions" feel predictable rather than familiar.

Assumption

Source ids deisenroth-2020-mml, goodfellow-2016-deep-learning must support the exact object, not just the surrounding topic.

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Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion.

Proposed experiment

Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo.

Next action

The learner can state the mechanism in their own words

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PredictionChecking carried observation

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AILearner handoff ready

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01PredictionChecking browser-local route memory

02EvidenceChecking for a carried observation

03BoundaryInspect source ids: deisenroth-2020-mml, goodfellow-2016-deep-learning

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Evidence to inspect

Source ids to inspect: deisenroth-2020-mml, goodfellow-2016-deep-learning
Definition, prerequisite, and contrast concept links
The equation or code witness that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Object-attached AI handoff

I am working in Continuous Function's research reading room. Object: concept - Distributions Object key: concept:probability/distributions Context: Probability Anchor id: concept/concept-notebook/probability/distributions Open question: What is the smallest example that makes Distributions click without losing the math? Evidence to inspect: - Source ids to inspect: deisenroth-2020-mml, goodfellow-2016-deep-learning - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids deisenroth-2020-mml, goodfellow-2016-deep-learning must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Distributions" feel predictable rather than familiar." | assumption: Source ids deisenroth-2020-mml, goodfellow-2016-deep-learning must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: deisenroth-2020-mml, goodfellow-2016-deep-learning" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Distributions" feel predictable rather than familiar. - Assumption to keep visible: Source ids deisenroth-2020-mml, goodfellow-2016-deep-learning must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

concept/concept-notebook/probability/distributions
concept:probability/distributions

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Reading map

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Maximum Likelihood

Connections and sources

Domain

Probability

probability
distributions
pmf
density

Prerequisites

Random Variables

Leads To

Maximum Likelihood

Object Companion

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A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take.

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Context prompt

You are my AI learning companion for Continuous Function. Current context: Probability concept. Learning surface: Distributions. What this page says: A distribution is the law of a random variable: it says how probability mass or density lands on the values the variable can take. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

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