{"id":"https://openalex.org/W7212010177","doi":"https://doi.org/10.48550/arxiv.2609.06859","title":"Normal random variables","display_name":"Normal random variables","publication_year":2026,"publication_date":"2026-09-06","ids":{"openalex":"https://openalex.org/W7212010177","doi":"https://doi.org/10.48550/arxiv.2609.06859"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.06859","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06859","pdf_url":"https://arxiv.org/pdf/2609.06859","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.06859","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025328470","display_name":"Teodor Banica","orcid":"https://orcid.org/0000-0003-4680-0364","observed_orcids":["https://orcid.org/0000-0003-4680-0364"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Banica, Teo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5025328470"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.7397000193595886,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.7397000193595886,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.2249000072479248,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.002899999963119626,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/normal-random-variables","display_name":"normal random variables","score":0.996999979019165},{"id":"https://openalex.org/keywords/central-limit-theorem","display_name":"central limit theorem","score":0.8450000286102295},{"id":"https://openalex.org/keywords/gaussian-random-vectors","display_name":"Gaussian random vectors","score":0.9490000009536743},{"id":"https://openalex.org/keywords/multivariate-normal-distribution","display_name":"multivariate normal distribution","score":0.7940000295639038},{"id":"https://openalex.org/keywords/free-probability","display_name":"free probability","score":0.7360000014305115},{"id":"https://openalex.org/keywords/random-matrices","display_name":"random matrices","score":0.949999988079071}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6248000264167786},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.6047999858856201},{"id":"https://openalex.org/C130648207","wikidata":"https://www.wikidata.org/wiki/Q6007562","display_name":"Imprecise probability","level":3,"score":0.599399983882904},{"id":"https://openalex.org/C89231458","wikidata":"https://www.wikidata.org/wiki/Q4723976","display_name":"Algebra of random variables","level":4,"score":0.5346999764442444},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.49410000443458557},{"id":"https://openalex.org/C102094743","wikidata":"https://www.wikidata.org/wiki/Q133871","display_name":"Normal distribution","level":2,"score":0.4212999939918518},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4066999852657318},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.3804999887943268},{"id":"https://openalex.org/C138405894","wikidata":"https://www.wikidata.org/wiki/Q3179949","display_name":"Multivariate random variable","level":3,"score":0.34779998660087585},{"id":"https://openalex.org/C182514268","wikidata":"https://www.wikidata.org/wiki/Q7636889","display_name":"Sum of normally distributed random variables","level":4,"score":0.34689998626708984},{"id":"https://openalex.org/C52386014","wikidata":"https://www.wikidata.org/wiki/Q166211","display_name":"Random element","level":3,"score":0.3450999855995178},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.3287999927997589},{"id":"https://openalex.org/C137002209","wikidata":"https://www.wikidata.org/wiki/Q898521","display_name":"Hidden variable theory","level":3,"score":0.31520000100135803},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3093000054359436},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3009999990463257},{"id":"https://openalex.org/C165770473","wikidata":"https://www.wikidata.org/wiki/Q6000597","display_name":"Illustration of the central limit theorem","level":5,"score":0.2858999967575073},{"id":"https://openalex.org/C187825244","wikidata":"https://www.wikidata.org/wiki/Q372071","display_name":"Probability-generating function","level":4,"score":0.2840000092983246},{"id":"https://openalex.org/C43596424","wikidata":"https://www.wikidata.org/wiki/Q98124957","display_name":"Probability and statistics","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C18653775","wikidata":"https://www.wikidata.org/wiki/Q1333358","display_name":"Joint probability distribution","level":2,"score":0.2741999924182892},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.273499995470047},{"id":"https://openalex.org/C165973109","wikidata":"https://www.wikidata.org/wiki/Q894014","display_name":"Quantum probability","level":5,"score":0.2623000144958496},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.25839999318122864},{"id":"https://openalex.org/C22679943","wikidata":"https://www.wikidata.org/wiki/Q159375","display_name":"Standard deviation","level":2,"score":0.25429999828338623},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.2502000033855438}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:2609.06859","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06859","pdf_url":"https://arxiv.org/pdf/2609.06859","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-05740196v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/hal-05740196v1","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"2026","raw_type":"info:eu-repo/semantics/preprint"},{"id":"doi:10.48550/arxiv.2609.06859","doi":"https://doi.org/10.48550/arxiv.2609.06859","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.06859","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.06859","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06859","pdf_url":"https://arxiv.org/pdf/2609.06859","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212010177.pdf","grobid_xml":"https://content.openalex.org/works/W7212010177.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7202293010","https://openalex.org/W1987591432","https://openalex.org/W4293195616","https://openalex.org/W2010961394","https://openalex.org/W487693173","https://openalex.org/W4238504555","https://openalex.org/W2807747831","https://openalex.org/W2385154303","https://openalex.org/W4232631157","https://openalex.org/W2251134411"],"abstract_inverted_index":{"This":[0],"is":[1],"an":[2,90],"advanced":[3],"introduction":[4,91],"to":[5,64,92,102],"the":[6,16,23,30,33,52,57,71,93,97],"various":[7],"types":[8],"of":[9,15,32,96],"normal":[10,36,54,98],"random":[11,109],"variables,":[12,37,55,99],"with":[13,38,67],"most":[14],"needed":[17],"preliminaries":[18],"included.":[19],"We":[20,61],"first":[21],"discuss":[22],"probability":[24,76,107],"basics,":[25],"standard":[26],"central":[27],"limits,":[28],"and":[29,41,56,74,82,84,100,108],"theory":[31],"usual,":[34],"real":[35],"examples,":[39],"illustrations":[40],"numerous":[42],"formulae.":[43],"Then":[44],"we":[45,88],"go":[46],"on":[47],"a":[48,68],"similar":[49],"discussion":[50,69],"regarding":[51,70],"complex":[53],"Rayleigh":[58],"variables":[59],"too.":[60,86],"then":[62],"move":[63],"arbitrary":[65],"dimensions,":[66],"Gaussian":[72],"vectors,":[73],"related":[75],"laws,":[77],"featuring":[78],"some":[79],"functional":[80],"analysis,":[81],"geometry":[83],"physics":[85],"Finally,":[87],"provide":[89],"quantum":[94],"versions":[95],"notably":[101],"those":[103],"coming":[104],"from":[105],"free":[106],"matrices.":[110]},"counts_by_year":[],"updated_date":"2026-10-27T15:51:10.671819","created_date":"2026-09-10T00:00:00"}
