{"doi":"10.1017/s0021900200018489","title":"On the nature of the binomial distribution","abstract":"<jats:p>We examine how the binomial distribution <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>) arises as the distribution <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>\n               </jats:sub> = ∑<jats:sub>\n                  <jats:italic>i</jats:italic>=1</jats:sub>\n               <jats:sup>\n                  <jats:italic>n</jats:italic>\n               </jats:sup> \n               <jats:italic>X</jats:italic>\n               <jats:sub>\n                  <jats:italic>i</jats:italic>\n               </jats:sub> of an arbitrary sequence of Bernoulli variables. It is shown that <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases <jats:italic>B</jats:italic>(2,<jats:italic>p</jats:italic>) and <jats:italic>B</jats:italic>(3,<jats:italic>p</jats:italic>) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>\n               </jats:sub> follows <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>), given that <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>-1</jats:sub> is not necessarily a binomial variable, are investigated. Several natural characterizations of <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.</jats:p>","journal":"Journal of Applied Probability","year":2001,"id":601380,"datarank":0.10397207708399181,"base_score":0.6931471805599453,"endowment":0.6931471805599453,"self_citation_contribution":0.10397207708399181,"citation_network_contribution":0.0,"self_endowment_contribution":0.10397207708399181,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":1,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":1542015,"name":"Abraham P. Punnen","orcid":null,"position":1,"is_corresponding":false},{"id":1542014,"name":"P. Vellaisamy","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"resolved":true,"title":"On the nature of the binomial distribution","abstract":"<jats:p>We examine how the binomial distribution <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>) arises as the distribution <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>\n               </jats:sub> = ∑<jats:sub>\n                  <jats:italic>i</jats:italic>=1</jats:sub>\n               <jats:sup>\n                  <jats:italic>n</jats:italic>\n               </jats:sup> \n               <jats:italic>X</jats:italic>\n               <jats:sub>\n                  <jats:italic>i</jats:italic>\n               </jats:sub> of an arbitrary sequence of Bernoulli variables. It is shown that <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases <jats:italic>B</jats:italic>(2,<jats:italic>p</jats:italic>) and <jats:italic>B</jats:italic>(3,<jats:italic>p</jats:italic>) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>\n               </jats:sub> follows <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>), given that <jats:italic>S</jats:italic>\n               <jats:sub>\n                  <jats:italic>n</jats:italic>-1</jats:sub> is not necessarily a binomial variable, are investigated. Several natural characterizations of <jats:italic>B</jats:italic>(<jats:italic>n</jats:italic>,<jats:italic>p</jats:italic>), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.</jats:p>","is_dataset_classified":null,"base_score":0.6931471805599453,"endowment":0.6931471805599453,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"23304386","pmcid":null,"openalex_id":"https://openalex.org/W4238600718","authors":[],"funders":[],"total_grants":0,"fwci":0.0,"citation_percentile":0.36686904,"influential_citations":0,"citation_trend":[],"oa_status":"closed","license":null,"oa_locations":[{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0021900200018489","host_type":"publisher"},{"url":"https://doi.org/10.1017/s0021900200018489","host_type":"journal"}],"fields_of_study":["Statistical Distribution Estimation and Applications","Fractional Differential Equations Solutions","Mathematical functions and polynomials"],"mesh_terms":[],"keywords":["Mathematics","Binomial distribution","Negative binomial distribution","Bernoulli distribution","Beta negative binomial distribution","Poisson binomial distribution","Beta-binomial distribution","Bernoulli's principle","Continuity correction","Binomial (polynomial)","Multinomial distribution","Poisson distribution","Distribution (mathematics)","Negative multinomial distribution","Binomial approximation","Random variable","Bernoulli process","Combinatorics","Statistics","Mathematical analysis","Physics"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-07-29T16:09:35.526126Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}