Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1205.0933

A note on the bivariate distribution representation of two perfectly correlated random variables by Dirac's $δ$-function

Andrés Alayón Glazunov·Jie Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we discuss the representation of the joint probability density function of perfectly correlated continuous random variables, i.e., with correlation coefficients $ρ=pm1$, by Dirac's $δ$-function. We also show how this representation allows to define Dirac's $δ$-function as the ratio between bivariate distributions and the marginal distribution in the limit $ρ\rightarrow \pm1$, whenever this limit exists. We illustrate this with the example of the bivariate Rice distribution

Explore related subjects

cs.NI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrés Alayón Glazunov, Jie Zhang. 2012-05-04. A note on the bivariate distribution representation of two perfectly correlated random variables by Dirac's $δ$-function. https://arxiv.org/abs/1205.0933

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Message-Level Scheduling for RLNC-Coded Multi-Source Traffic

This paper studies weighted decoding-delay minimization for multiple RLNC-coded message streams that compete for finite processing capacity at a destination. Packet arrivals are exogenous, while the scheduler only determines the processing order of packets already available at the destination. A trace-conditioned offline scheduling formulation shows that a batch-release subclass is strongly NP-hard even with a single processing unit. Message-Aware Innovation-Deficit Scheduling (MAIDS) is then developed to prioritize each serviceable message according to its weight and remaining decoding deficit. For a single processing unit, MAIDS is shown to be exactly optimal under nonblocking progressive arrivals with equal weights and under common activation with arbitrary positive weights, while the unrestricted weighted online problem admits no universal deterministic $O(1)$ competitive ratio. Simulation results on streaming and batch benchmarks show that MAIDS consistently reduces weighted decoding delay relative to the tested baselines, remains close to the offline optimum on average, and recovers the predicted exact performance boundaries.

cs.NI↗
arXiv2026

The Towers Were Standing: A Cause Decomposition of Cellular Outages During Hurricane Helene

Hurricane Helene produced the largest absolute cell-site outage in the public FCC record, peaking at 4562 sites. The conventional model is physical: towers destroyed. Helene did destroy over 1700 miles of fibre, but almost none of it was cell sites. We present the first cause-decomposed study of the FCC's Disaster Information Reporting System, reconstructing 80 state-days and 580 county-days from 24 daily filings by two reconciled independent extractions. Damage to cell sites is negligible: 1.1% of attributed cell-site-days across six states, at most 3.8% anywhere. The sites were standing. What took them out divides by terrain: pooled, power dominates at 63.2%, but in mountainous North Carolina severed transport (backhaul) reaches 52.2% against 47.3%, and in Tennessee 69.9%. North Carolina's transport share rises from 7.0% to 85.0% across the event (\r{ho} = 0.92). Seventeen days after landfall, on 15 October, 47 sites lost transport across six contiguous North Carolina counties with no rainfall, no power loss, no damage, and recovery by the next report. Independent active-probe measurement corroborates it: responsive /24s fall 1.02% for twelve hours while Tennessee stays flat. We release the dataset. Backup power is the standard resilience investment; here it addresses the smaller half of the problem.

cs.NI↗
arXiv2026

terms.txt: A Consent and Compensation Protocol for Agentic Web Access

The open web ran on an unwritten bargain: sites admitted crawlers, and search engines sent visitors back. Public measurements show that bargain breaking under AI crawlers and agents. Automated clients now make up most requests, training dominates Cloudflare-classified crawling, and the largest AI platforms fetch thousands of pages for each visitor they return. The web's common control, robots.txt, cannot express identity, purpose, terms, or price, can be circumvented, and newer alternatives are largely proprietary CDN features. We specify terms.txt, a robots.txt-style file for per-path, per-purpose machine-access terms, plus an origin-enforced exchange using Web Bot Auth signatures, signed intent, delegation tokens, HTTP 402 negotiation, and signed receipts. We define what the exchange can enforce, audit, and leave to contract. A dependency-free implementation adds 0.20 to 0.65 ms per request on one vCPU.

cs.NI↗