LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Multinomial mass function is normalized

Statement

For $k \geq 1$ , $n \in N_{0}$ , and $p_{j} \geq 0$ with $\sum_{j} p_{j} = 1$ , the masses $\frac{n!}{\prod_{j = 1}^{k} y_{j}!} \prod_{j = 1}^{k} p_{j}^{y_{j}}, y_{j} \in N_{0}, \sum_{j} y_{j} = n,$ sum to one, using $0^{0} = 1$ .

Facts & Assumptions

Given: The finite probability vector $p$ , $k \geq 1$ , and $n \in N_{0}$ .

[F1]

The multinomial expansion is $(\sum_{j} p_{j})^{n} = \sum_{\sum y_{j} = n} n! \prod_{j} p_{j}^{y_{j}} / \prod_{j} y_{j}!$ . The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ .

[F2]

For nonnegative integer counts summing to $n$ , the multinomial coefficient defined as a coloring count equals $n! / \prod_{j} y_{j}!$ . The multinomial coefficient $(\binom{n}{k_{0}, \dots, k_{m - 1}})$ as the number of ordered partitions of an $n$ -set into blocks of prescribed sizes The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ .

Proof

Given: $p_{j} \geq 0$ , $\sum_{j} p_{j} = 1$ , $k \geq 1$ , and $n \in N_{0}$ .

1.1givenF1F2algebra

Apply [F1] and [F2] to expand $(\sum_{j} p_{j})^{n}$ ; each term is exactly the stated nonnegative mass. Since $\sum_{j} p_{j} = 1$ , the sum of all masses is one.

2.1step 1.1givencases∎

When $n = 0$ there is one all-zero count vector and its mass is $0! / (0!)^{k}$ times a product of $k$ zero-th powers, hence one. When some $p_{j} = 0$ , terms with $y_{j} > 0$ vanish and terms with $y_{j} = 0$ use $0^{0} = 1$ ; the same expansion remains valid.

Depends on

Used by

Multinomial distribution Definition
Dirichlet-Multinomial conjugate update Theorem

Dependency tree · two levels

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Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §3.4 (standard reference, not scraped)
Gero Walter, A Technical Note on the Dirichlet-Multinomial Model, 2013 (standard reference, not scraped)