LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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 Binomial count has its usual probability mass function

Statement

If $X_{1}, \dots, X_{n}$ are independent Bernoulli $(p)$ variables with $0 < p < 1$ and $n \in N_{0}$ , then their sum $S_{n}$ has $P (S_{n} = y) = (\binom{n}{y}) p^{y} (1 - p)^{n - y}, y = 0, \dots, n .$ For $n = 0$ , the empty sum is zero and the sole mass is one.

Facts & Assumptions

Given: Independent Bernoulli $(p)$ variables, $0 < p < 1$ , and $n \in N_{0}$ .

[F1]

A Binomial $(n, p)$ variable is the sum of $n$ independent Bernoulli $(p)$ variables. Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials.

[F2]

The binomial coefficient $(\binom{n}{y})$ counts the $y$ -element subsets of an $n$ -element set. The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ .

[F3]

Independent random elements have the product joint law. Independent random elements have product joint law.

Proof

Given: The hypotheses in the Statement.

1.1givenF2F3cases

For $n \geq 1$ , each particular subset $J \subseteq {1, \dots, n}$ of size $y$ describes the disjoint event that $X_{i} = 1$ exactly for $i \in J$ . Independence [F3] gives this event probability $p^{y} (1 - p)^{n - y}$ . There are $(\binom{n}{y})$ such subsets by [F2], and their union is ${S_{n} = y}$ .

2.1step 1.1F1algebra∎

Summing the equal probabilities in step 1.1 gives the formula. If $n = 0$ , the sum is deterministically zero and the empty product is one, which gives the stated boundary case as well.

Depends on

Used by

Beta-Binomial conjugate update Theorem

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §2.1 (standard reference, not scraped)
Johnson, Ott & Dogucu, Bayes Rules!, Chapter 3, §3.1 (standard reference, not scraped)