bibkey: siegrist2026generating authors: Kyle Siegrist year: 2026 title: “Random: Probability, Mathematical Statistics, Stochastic Processes” doi: null url: https://www.randomservices.org/random/ claim: The probability generating function gives P(N even)=(1+G(-1))/2; for a binomial sum of d independent Bernoulli(p) bits, G(t)=(1-p+pt)^d, and conditioning on a positive-probability event defines a probability law. strata_touched:

• D5/S3/TotalVariation/ParityKernelMasses
• D5/S3/TotalVariation/TreeParityKernel license: CC-BY-2.0 triage: anchor

Bernoulli parity and conditional normalization

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The source is Kyle Siegrist’s online textbook Random: Probability, Mathematical Statistics, Stochastic Processes. The bibliographic year identifies the consulted online version; the pages do not state a publication year. Its home page specifies CC BY 2.0 and requires attribution and a link to the home site. No DOI is supplied by the source.

The following named sections and anchored statements supply the identities:

• Generating Functions, The Probability Generating Function, pgf4 states P(N is even) = (1 + G(-1))/2 and proves it by adding G(1) and G(-1).
• The Binomial Distribution, Moments, mom3 states G(t) = (1-p+pt)^d for the sum of d independent Bernoulli(p) bits, for every real t.
• Conditional Probability, dfn defines P(A | B) = P(A intersect B)/P(B) when P(B) > 0; its Basic Theory also proves that this conditional law is a probability measure.

These HTML statements, their proofs, the author metadata and the home-page license identify the source and are the verification method for this note. The mathematical descriptions here are adapted from that source with attribution.

For a bit vector x, let h(x) count its occupied coordinates. The even formula and its complementary odd formula give

$$\mathbb{P}(h(X)\equiv M(\mathrm{mod}2))=\frac{1+(-1{)}^M(1-2p{)}^d}{2}.$$

For positive natural d and M, take p = M/(2M+d). Then 1-2p = d/(2M+d) lies strictly between zero and one, so the displayed probability p_e is positive. The vector mass conditioned on this event is

$$Q(x)=\{\begin{array}{ll}{p}^{h(x)}(1-p{)}^{d-h(x)}/p_e,& h(x)\equiv M(\mathrm{mod}2),\\ 0,& \text{otherwise}.\end{array}$$

It is nonnegative and sums to one by the cited conditional-probability law. This is the textbook identity at the displayed parameter choice, not a claim that the conditioned coordinates remain independent. The source does not identify this law with an actual ordered-tree parity law or supply the finite total-variation comparison between that law and uniform weak compositions.