Technical & Instructional Articles The Rational Gambler (2010) Introducing the concept of the "rational gambler" and gambling as an optimization of variance versus expected loss Craps: Computing the Distribution of the Pass-Line and Free Odds Bets (2010) Understanding the statistical properties of the payoffs from these craps bets Implying the Risk-Neutral Distribution from the Volatility Skew in Options Pricing (2008) Relating the probability distribution of the underlying to the Black-Scholes implied volatility skew Probability Density Function in Terms of Moments (2008) A closed-form expression for the PDF of a random variable in terms of its moments The Cereal Box Problem (2007) A probability brainteaser The Musical Chairs Problem (2007) A probability brainteaser Distribution of R2 for Single Regression of Uncorrelated Gaussian Random Variables (2007) A closed-form expression for the probability density function of the coefficient of determination in this special case Moments of the Standard Normal Probability Density Function (2007) A closed-form expression for the moments of the standard normal Markowitz Efficient Portfolio Frontier as Least-Norm Analytic Solution to Underdetermined Equations (2007) A closed-form quadratic equation for the Markowitz efficient frontier in modern portfolio theory Estimating Distribution Parameters using Coarse Data for Chi-Squared Goodness-of-Fit Test (2007) Two methods for estimating the parameters of the null distribution using coarse data Proof that the Difference of Jointly Distributed Normal Random Variables is Normal (2007) An exercise in calculus The Number of Digits in n! (2007) A fun investigation in number theory Proof that Superposition Implies Rational Homogeneity (2007) A nice result from linear systems theory Expected Value of the Rayleigh Random Variable (2006) An entertaining integral Closed-Form Solution to the Lights Out Puzzle (2006) A mathematical investigation of a child's toy Derivation of the Matched Filter as the Highest SNR Linear Estimator (2006) I've also contributed this derivation to Wikipedia. An Intuitive Approach to Chemical Equilibrium (2006) A short supplementary text on chemical equilibrium illustrating some basic numerical methods for solving equilibrium systems Derivation of Constant Labor and Capital Share from the Cobb-Douglas Production Function (2006) Taken from the back of an envelope Integration by Parts: An Intuitive and Geometric Explanation (2005) A visual alternative to the derivation of the integration by parts formula The Rabbani-Kloek Elimination (2005) A method for answering multiple choice questions on an AP exam Patents, Publications, and Research S. Rabbani & H. Shahmohamad. "Statistical Brainteasers in Algorithms, Counting & Strategy." Open Journal of Statistics. Vol. 4, No. 4, pp. 278-283 (2014). S. Rabbani & H. Shahmohamad. "Probability Brainteasers for College Students." Open Journal of Statistics. Vol. 4, No. 4, pp. 273-277 (2014). P. Paul & S. Rabbani. "Banding Profile Estimator Using Multiple Sampling Intervals." United States Patent No. 7,855,806. Issued 21 December 2010. P. Paul, S. Rabbani, & W. J. Nowak. "Method and System to Compensate for Banding Defects." United States Patent No. 7,755,799. Issued 13 July 2010. A. P. T. Lau, S. Rabbani, & J. M. Kahn. "On the Statistics of Intra-Channel Four-Wave Mixing in Phase-Modulated Optical Communication Systems." IEEE Journal of Lightwave Technology. Vol. 26, Issue 14, pp. 2128-2135 (2008). Travel Letters From the Baltics (2023) Helsinki, Finland; Tallinn/Pärnu, Estonia; Riga/Tērvete, Latvia; Šiauliai/Kaunas/Vilnius, Lithuania (28 June 2023 - 8 July 2023) From Panamá (Boquete, Panama City, Fort Sherman, Gamboa) (2023) Boquete, Chiriquí, Panamá; Panama City, Panamá, Panamá; Fort Sherman, Colón, Panamá; Gamboa, Colón, Panamá (14 January 2023 - 21 January 2023) From Spain (Madrid, Sevilla, Galicia, and Jiménez de Jamuz) (2022) Madrid, Comunidad de Madrid, Spain; Sevilla, Andalucía, Spain; Santiago de Compostela/A Coruña/Cambados/O Grove/Illa da Toxa/Vigo/Illa de Arousa/Combarro, Galicia, Spain; Jiménez de Jamuz, Castilla y León, Spain (28 May 2022 - 8 June 2022) From New Zealand (2019) Auckland/Rangitoto/Waitomo/Taupo/Tongariro/Napier/Wellington/Queenstown/Te Anau/Milford Sound/Wanaka/Aoraki Mt. Cook/Tekapo/Christchurch, New Zealand (17 November 2019 - 28 November 2019) From Trieste and Istria (2019) Staro Selo, Slovenia; Trieste, Italia; Motovun/Rovinj, Croatia (5 July 2019 - 7 July 2019) From Doha (2019) Doha, Qatar (6 May 2019 - 7 May 2019) From the Caucasus (2019) Yerevan/Halidzor/Goris/Noravank/Sevan/Dzoraget/Alaverdi, Armenia; Tbilisi/Mtskheta, Georgia; Baku/Qobustan, Azerbaijan (26 April 2019 - 6 May 2019) From Bangkok (2018) Bangkok, Thailand (16 November 2018 - 21 November 2018) From India (2018) Hyderabad/Kochi/Alappuzha/Munnar/Jaipur/Agra/Delhi, India (11 April 2018 - 22 April 2018) From Costa Rica (2018) Grecia/La Fortuna/Arenal/Monteverde/Sarchi, Costa Rica (19 January 2018 - 22 January 2018) From Yangon (2017) Yangon, Myanmar (1 September 2017 - 5 September 2017) From Norway (2017) Oslo/Myrdal/Flåm/Gudvangen/Bergen/Stavanger, Norway (25 June 2017 - 2 July 2017) From Japan (2017) Tokyo/Kamakura/Kyoto/Nara/Osaka/Hiroshima/Kobe, Japan (23 February 2017 - 5 March 2017) From Kuala Lumpur (2016) Kuala Lumpur, Malaysia (29 July 2016 - 31 July 2016) From Vancouver (2016) Vancouver, British Columbia, Canada (30 June 2016 - 3 July 2016) From Italia II (Marche, Emilia-Romagna, Toscana, Umbria, Lazio) and San Marino (2016) Sforzacosta/Macerata/Offida/Ascoli Piceno/Ancona/Gradara/Bologna/Siena/Montalcino/San Quirico d'Orcia/Pienza/Montepulciano/Perugia/Spello/Assisi/Orvieto/Roma, Italia; Murata/Città di San Marino/Borgo Maggiore, San Marino (8 April 2016 - 17 April 2016) From México (México, D.F. II, Teotihuacan, Puebla, Chipilo, Cholula) (2015) México, D.F./Teotihuacan/Puebla/Chipilo/Cholula, México (4 October 2015 - 11 October 2015) From Singapore (2015) Singapore (1 May 2015 - 4 May 2015) From the Netherlands (Voorburg, Delft, Rotterdam, The Hague, Amsterdam) (2014) Voorburg/Delft/Rotterdam/The Hague/Amsterdam, the Netherlands (13 December 2014 - 15 December 2014) From Italia (Roma, Vaticano, Costiera Amalfitana, Napoli, Ercolano) (2014) Roma/Vaticano/Sorrento/Amalfi/Ravello/Positano/Napoli/Ercolano, Italia (14 March 2014 - 24 March 2014) From Berlin II (2014) Berlin, Germany (14 March 2014) From Zürich (2014) Zürich, Switzerland (1 February 2014 - 3 February 2014) From Barcelona (2013) Barcelona, Spain (21 September 2013 - 24 September 2013) From Hong Kong and Macau (2013) Hong Kong (SAR), China; Macau (SAR), China (13 July 2013 - 22 July 2013) From Copenhagen and Helsinki (2013) Copenhagen, Denmark; Helsinki, Finland (4 July 2013 - 9 July 2013) From Montréal (2013) Montréal, Québec, Canada (31 May 2013 - 2 June 2013) From México (San Miguel de Allende, México, D.F.) (2013) San Miguel de Allende/México, D.F., México (25 May 2013 - 29 May 2013) From Southeast Asia (Thailand, Cambodia, Laos, Vietnam) (2012) Bangkok/Ayutthaya, Thailand; Phnom Penh/Siem Reap/Angkor Wat/Kampong Phluk, Cambodia; Luang Prabang, Laos; Hanoi/Saigon, Vietnam (10 November 2012 - 22 November 2012) From the Basque Country, Spain/France (Bilbao, Getxo, San Sebastián, Hondarribia, Saint-Jean-de-Luz, Biarritz, Pamplona) (2012) Bilbao/Getxo/San Sebastián/Hondarribia/Pamplona, Spain; Saint-Jean-de-Luz/Biarritz, France (10 March 2012 - 17 March 2012) From Malta (2011) Malta and Gozo, Malta (26 August 2011 - 30 August 2011) From Lisbon (2011) Lisbon and Sintra, Portugal (19 August 2011 - 21 August 2011) From Iceland (2011) Western and southern coasts and West Fjords, Iceland (1 July 2011 - 10 July 2011) From Ireland (Cork, Kinsale, Dublin) (2011) Cork/Kinsale/Dublin, Ireland (29 April 2011 - 1 May 2011) From Hungary (Budapest II), Slovakia (Vysoké Tatry), Poland (Kraków, Auschwitz, Warszawa) (2010) Budapest, Hungary; Vysoké Tatry, Slovakia; Kraków/Auschwitz/Warszawa, Poland (9 October 2010 - 19 October 2010) From China (Shanghai, Tunxi, Huangshan) (2010) Shanghai/Tunxi/Huangshan, China (6 July 2010 - 16 July 2010) From Egypt (Alexandria, Luxor, Aswan, Cairo) (2009) Alexandria/Luxor/Aswan/Cairo, Egypt (14 February 2009 - 25 February 2009) From Sicilia, Italia (Palermo, Agrigento, Cefalù, Catania, Taormina, Enna) and Slovenia (Ljubljana, Bled) (2008) Sicily, Italy; Ljubljana/Bled, Slovenia (1 June 2008 - 14 June 2008) From Campania, Italia (Napoli, Monte Vesuvio, Pompei) (2008) Campania, Italy (29 May 2008 - 1 June 2008) From Rome (2008) Rome, Italy (27 May 2008 - 29 May 2008) From Serbia (Belgrade), Macedonia (Skopje), Greece (Thessaloniki, Athens) (2008) Belgrade, Serbia; Skopje, Macedonia; Thessaloniki/Athens, Greece (22 May 2008 - 27 May 2008) From Albania (Shkodër, Tiranë, Vuno, Vlorë, Berat, Bajram Curri) and Kosovo (Drakovica, Priština) (2008) Albania; Kosovo (15 May 2008 - 22 May 2008) From Croatia (Zagreb, Split, Dubrovnik), Bosnia and Herzegovina (Reum, Kravice, Mostar, Blagaj, Pocitelj), Montenegro (Budva, Ulcinj); Introduction to Albania (2008) Croatia; Bosnia and Herzegovina; Montenegro (8 May 2008 - 15 May 2008) From Budapest (2008) Budapest, Hungary (6 May 2008 - 8 May 2008) From Romania (Bucharest, Constanta, Brasov, Cluj-Napoca) (2008) Romania (27 April 2008 - 3 May 2008) From Sofia (2008) Sofia, Bulgaria (24 April 2008 - 27 April 2008) From Istanbul (2008) Istanbul, Turkey (21 April 2008 - 24 April 2008) From Austria (Vienna), Slovakia (Bratislava), Germany (Berlin) (2008) Vienna, Austria; Bratislava, Slovakia; Berlin, Germany (9 April 2008 - 18 April 2008) From Prague (2007) Prague, Czech Republic (5 April 2007 - 11 April 2007) Translations, Poetry, & Essays Translations of Selected Quatrains from Khayyám's Rubáiyát from Persian (2001-2024) An ongoing translation project that attempts to preserve both rhyme and meaning Traduko de teksto de Radiohead de angla el esperanto (2005) For the Esperantistoj out there The Rights of the People, Denied and Disparaged (2005) A closer look at the Ninth Amendment of the U.S. Constitution Translations of Selected Lyrics of Persone from Esperanto (2004) Persone is an Esperanto rock band based in Sweden. Cherry Blossom Petals and Camera Bags (2004) From Poems from Kyoto The Last Train to Finchley Road (2004) From Poems from Kyoto Ode on a Perrier Bottle (2004) Not a Grecian urn Copyright © 2025 Sahand Rabbani All rights reserved.