{"id":2301,"date":"2025-08-18T16:53:05","date_gmt":"2025-08-18T23:53:05","guid":{"rendered":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/?page_id=2301"},"modified":"2025-11-08T23:29:36","modified_gmt":"2025-11-09T07:29:36","slug":"math-505a-applied-probability","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-505a-applied-probability\/","title":{"rendered":"Math 505a: Applied Probability"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <p>Everything can change with little or no notice at any moment\u2026<\/p>\n<p><strong>In particular, a\u00a0 lecture\u00a0 can be moved to on-line mode on a very short notice, so please check your e-mail before every class.<\/strong><\/p>\n<h1 align=\"center\">Class number 39710<\/h1>\n<p><span style=\"color: #ff0000;\">Math 505a in Fall 2025 semester: Key dates<\/span><\/p>\n<ul>\n<li>August 25: first day of classes<\/li>\n<li>September 1: Labor Day, no class<\/li>\n<li>September 12: Last day to drop without a <em><strong>W<\/strong><\/em> AND with refund<\/li>\n<li>October 6: Midterm Exam 1<\/li>\n<li>October 9,10: Fall Break<\/li>\n<li>October 10: Last day to drop without a <em><strong>W<\/strong><\/em>, BUT WITH NO refund<\/li>\n<li>November 14: Last day to drop with a <em><strong>W<\/strong><\/em><\/li>\n<li>November 17: Midterm Exam 2<\/li>\n<li>November 26-30: Thanksgiving Break<\/li>\n<li>December 5: Final project is due; Last day of classes<\/li>\n<li>December 17: Final exam (11am-1pm, in KAP 146)<\/li>\n<\/ul>\n<h3 style=\"text-align: center;\"><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Sched-505a-F2025.pdf\" target=\"_blank\" rel=\"noopener\">Class schedule<\/a><\/h3>\n<h3 style=\"text-align: center;\"><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/HW-505a-F2025.pdf\" target=\"_blank\" rel=\"noopener\">Homework problems and the project<\/a><\/h3>\n<p>&nbsp;<\/p>\n<p><strong>Instructor: <\/strong> Dr. Sergey Lototsky<br \/>\n<strong> Office:<\/strong> KAP 248D.<br \/>\n<strong> Phone:<\/strong> 213\u2013740-2389.<br \/>\n<strong> E-mail:<\/strong> lototsky (at) USC (dot) edu<br \/>\n<strong> URL:<\/strong> https:\/\/dornsife.usc.edu\/sergey-lototsky\/<br \/>\n<strong>Lectures:\u00a0 <\/strong>MWF 1-1:50pm, KAP 146<br \/>\n<strong>Office hours: <\/strong>MWF 2:15-3:15pm [in KAP 248D]<\/p>\n<h3><span style=\"color: #339966;\">Please do not hesitate to talk to me about your problems, questions, or concerns in this class. We can always arrange a special zoom meeting.<\/span><\/h3>\n<p><strong>Teaching Assistant (Grader):<\/strong> <a href=\"https:\/\/sites.google.com\/usc.edu\/inga-girshfeld\/\" target=\"_blank\" rel=\"noopener\"><span data-olk-copy-source=\"MessageBody\">Inga Girshfeld<\/span><\/a><br \/>\n<strong>E-mail:<\/strong> <span class=\"OZZZK\">girshfel {at} usc [dot] edu<\/span><br \/>\n<strong>Office hours:<\/strong> Tu 1-2pm, W 2-4pm, all in the Math Center (KAP 263).<\/p>\n<p><strong>Textbook:<\/strong> Probability and Random Processes by Geoffrey R. Grimmett and David R. Stirzaker Oxford University Press. 4th Edition (2020) is the most recent; ISBN 978\u20130\u201319\u2013884760\u20131 (Hardcover) ISBN 978\u20130\u201319\u2013884759\u20135 (Paperback)<\/p>\n<p><strong>Objective:<\/strong> To gain general knowledge and skills necessary to apply probabilistic methods to various areas of natural and social sciences.<\/p>\n<p><strong>Goal:<\/strong> To be ready for Math 505b (or its equivalent)<\/p>\n<p><strong>Concrete task:<\/strong> To be ready for a PhD-level comprehensive exam in (non-measure-theoretic) probability.<\/p>\n<p><span style=\"color: #00ffff;\"><span style=\"color: #339966;\"><em>Note:<\/em>\u00a0 In our department (USC Mathematics), the graduate probability exam based on the material covered in MATH 505a is <strong>required<\/strong> for Ph.D. in Applied Mathematics\u00a0 and is an <strong>option<\/strong> for Ph.D. in Mathematics, Masters in Applied Mathematics (both MA and MS), and Masters in Statistics.\u00a0<\/span> \u00a0<\/span><\/p>\n<h3><strong>Save the dates! <\/strong><span style=\"color: #ff0000;\"> There will be two in-class one-hour exams: Monday, October 6 and Monday,\u00a0 November 17, during regular lecture hours, in the regular lecture room (KAP 146). Final exam is Wednesday, December 17, 11am-1pm, in the regular lecture room.<\/span><\/h3>\n<p><strong>Note:<\/strong> You might need a PDF Scanner, to submit your (mostly likely handwritten) work to Gradescope<br \/>\nin PDF format (one PDF per assignment or exam). There are many free phone apps that you can use<br \/>\nto scan your work to make a PDF, for instance Adobe Scan.<\/p>\n<p><strong>Homework:<\/strong> There will be six homeworks and a final project. You should know how to solve every<br \/>\nhomework problem and turn in each homework on the corresponding due date. You are welcome to use<br \/>\nany help with all the work you do outside the classroom. During exams, you are on your own. No<br \/>\ncalculators will be necessary (or allowed) during exams. All exams are closed book.<\/p>\n<p>Please keep in mind that homework assignments are minimal requirements. To succeed in the class,<br \/>\nyou need to solve more problems, from the book and\/or from other sources. <span style=\"color: #ff0000;\">Keep all your notes,<\/span><br \/>\n<span style=\"color: #ff0000;\">including scratch paper, until after you are completely done with this class (or with the corresponding Graduate Exam<\/span><span style=\"color: #ff0000;\">).<\/span><\/p>\n<p><strong>Note on Use of AI<\/strong><br \/>\nYou are welcome to use\u00a0 artificial intelligence (AI) powered programs as a help with homework problems, but <em>not with\u00a0 exams<\/em>. While AI tools can help you brainstorm ideas or revise work you have already written, AI text generation tools may present incorrect information, biased responses, and incomplete analyses. To adhere to our university values, you must cite any AI-generated material (such as text or images) included or referenced in your work and provide the prompts used to generate the content.\u00a0 I will not be using any AI tools in grading your work.<\/p>\n<p><strong>Grading:<\/strong><\/p>\n<ul>\n<li>Homeworks 20% total<\/li>\n<li>Two One-Hour Exams, 40% total (20% each)<\/li>\n<li>Final Project 10%<\/li>\n<li>Final Exam 30%<\/li>\n<\/ul>\n<p><span style=\"color: #000000;\"><strong>To put is differently:<\/strong><\/span> you get 70% of the final grade from in-class work (two midterms and the final<br \/>\nexam) and 30% of the final grade from homeworks and the project.<\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Topics-505a-par.pdf\" target=\"_blank\" rel=\"noopener\">List of topics to be learned<\/a>\u00a0 \u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Topics-505a-refs.pdf\" target=\"_blank\" rel=\"noopener\">More detailed version with references<\/a><\/p>\n<p><strong>Previous Graduate Exams on the topic of the class from the USC Math Department:<\/strong><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Math505a_Pre_2012.pdf\" target=\"_blank\" rel=\"noopener\">1999-2011<\/a>\u00a0 \u00a0<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Math505a_2012-2025.pdf\" target=\"_blank\" rel=\"noopener\">2012-2025<\/a>\u00a0 \u00a0<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/10\/505aFall2025.pdf\">505aFall2025<\/a><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/mgsa\/qualifying-exams\/applied-probability\/\" target=\"_blank\" rel=\"noopener\">Some ideas for some of the solutions<\/a><\/p>\n<p><strong> Other materials <\/strong><\/p>\n<ul>\n<li><strong> Mine<\/strong>\n<ul>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/GeneralIntro-Ugrad-ProbabStat.pdf\" target=\"_blank\" rel=\"noopener\">Lecture 1<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/11\/Counting.pdf\" target=\"_blank\" rel=\"noopener\">A summary of combinatorics<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Counting-Examples.pdf\" target=\"_blank\" rel=\"noopener\">Some basic counting problems<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/CountingTriangles.jpg.jpg\" target=\"_blank\" rel=\"noopener\">Counting triangles<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/GiftsToKids.pdf\" target=\"_blank\" rel=\"noopener\">\u201cGifts to kids\u201d problem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FiveProbabProblems.pdf\" target=\"_blank\" rel=\"noopener\">Some famous problems in probability<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/09\/ODEvsFDE.png\" target=\"_blank\" rel=\"noopener\">2nd order linear constant coefficients:\u00a0 ODEs vs FDEs<\/a><\/li>\n<li>Random variables: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RandomVariables-General.pdf\" target=\"_blank\" rel=\"noopener\">general definitions<\/a> and an <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RandVar-Pics.pdf\" target=\"_blank\" rel=\"noopener\">easy diagram<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/05\/DiscrRandVar-new.pdf\" target=\"_blank\" rel=\"noopener\">A summary of discrete random variables<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/BaselAndGauss.pdf\" target=\"_blank\" rel=\"noopener\">Two computations: the Basel problem and the Gaussian integral<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Normal-History.pdf\" target=\"_blank\" rel=\"noopener\">Gaussian distribution: a time line<\/a><\/li>\n<li>Normal approximation: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/09\/Normal-binomHalf.pdf\" target=\"_blank\" rel=\"noopener\">Binomial(N=30, p=0.5)<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/09\/Normal-binom-one-tenth.pdf\" target=\"_blank\" rel=\"noopener\">Binomial(N=30, p=0.1)<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/09\/Normal-Poisson36point6.pdf\" target=\"_blank\" rel=\"noopener\">Poisson(36.6)<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/09\/MatLabForNormalApproximation.pdf\" target=\"_blank\" rel=\"noopener\">MatLab codes<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Normal-summary.pdf\" target=\"_blank\" rel=\"noopener\">Gaussian objects: Normal random variables, CLT, and more<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/07\/CauchyDistribution.pdf\" target=\"_blank\" rel=\"noopener\">Cauchy distribution<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/NormalTailBounds.pdf\" target=\"_blank\" rel=\"noopener\">The normal tail<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/08\/GammaBetaFunctions.pdf\" target=\"_blank\" rel=\"noopener\">Gamma and Beta Functions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/CharFunction.pdf\" target=\"_blank\" rel=\"noopener\">A summary of characteristic functions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/HarmonicNumbers.pdf\" target=\"_blank\" rel=\"noopener\">About harmonic numbers<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Inequalities-Probab-Basic.pdf\" target=\"_blank\" rel=\"noopener\">Basic inequalities<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/05\/Inequalities.png\" target=\"_blank\" rel=\"noopener\">More on probability inequalities<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/RV-Convergence.pdf\" target=\"_blank\" rel=\"noopener\">Convergence of random variables<\/a> and an<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Convergence-Pics.pdf\" target=\"_blank\" rel=\"noopener\"> illustration<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PoissonArrivals.png\" target=\"_blank\" rel=\"noopener\">Arrivals in the Poisson process<\/a> and <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Uniform.pdf\" target=\"_blank\" rel=\"noopener\">an illustration of clustering<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SSRW-onR-summary.pdf\" target=\"_blank\" rel=\"noopener\">A summary of simple symmetric random walk on R<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RandomGeneration.pdf\" target=\"_blank\" rel=\"noopener\">A summary of random object generation<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/BuffonNeedleAndMore.pdf\" target=\"_blank\" rel=\"noopener\">Buffon\u2019s needle and more<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/WeierApprTh.pdf\" target=\"_blank\" rel=\"noopener\">The Weierstrass (polynomial) approximation theorem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SummaryOfProbability-2.pdf\" target=\"_blank\" rel=\"noopener\">A summary of undergraduate probability<\/a> (for example USC MATH 407)<\/li>\n<\/ul>\n<\/li>\n<li><strong>Found on line<\/strong>\n<ul>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Bernoulli_triangle_columns.png\" target=\"_blank\" rel=\"noopener\">When the obvious pattern breaks<\/a> and the <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Dividing-a-circle-into-areas.pdf\" target=\"_blank\" rel=\"noopener\">related math story<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/VennDiagramsAndMore.pdf\" target=\"_blank\" rel=\"noopener\">Venn diagrams and other nice pictures<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FubiniNumbers-Asymptotics-short.pdf\" target=\"_blank\" rel=\"noopener\">Ordered partitions of a set<\/a>\u00a0 (asymptotic of the Fubini numbers)<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SetPartitionsdiBruno.pdf\" target=\"_blank\" rel=\"noopener\">Partitions and the chain rule from calculus<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/CatalanNumbers1.pdf\" target=\"_blank\" rel=\"noopener\">About Catalan numbers<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/ProbabDistributions.pdf\" target=\"_blank\" rel=\"noopener\">(Almost) everything you need to know about probability distributions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PoissonRVandProcess-Arratia.pdf\" target=\"_blank\" rel=\"noopener\">Poisson distribution and process by R. Arratia<\/a><\/li>\n<li>Buffon\u2019s needle: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/BuffonsNeedlingAnts.pdf\" target=\"_blank\" rel=\"noopener\">an application<\/a> and a <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Lazzarini.pdf\" target=\"_blank\" rel=\"noopener\">story<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Sylvester-4points.pdf\" target=\"_blank\" rel=\"noopener\">Sylvester\u2019s four point problem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/09\/Coincidences.pdf\" target=\"_blank\" rel=\"noopener\">Beyond the standard birthday problem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/OccupancyProblem.pdf\" target=\"_blank\" rel=\"noopener\">Occupancy problems<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/CouponCollectorProblem-UCDavis.pdf\" target=\"_blank\" rel=\"noopener\">Coupon collection problem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/BenfordLaw-basic.pdf\" target=\"_blank\" rel=\"noopener\">Benford\u2019s Law<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/NewtonDeSalvo-Sudoku.pdf\" target=\"_blank\" rel=\"noopener\">A research paper on random sudoku tables<\/a> [by USC people]<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RelPrime-ResearchPaper.pdf\" target=\"_blank\" rel=\"noopener\">A research paper on probability and number theory<\/a>\u00a0 [more than two relatively prime numbers]<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SizeBiasing-Survey.pdf\" target=\"_blank\" rel=\"noopener\">Size-biasing<\/a> [a survey paper by USC Math Professors]<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/UniformOnTheSphere.pdf\" target=\"_blank\" rel=\"noopener\">Generating a uniform distribution on the sphere<\/a><\/li>\n<li>For even more practice, here are old statistics exams based on Math 541ab: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/StatA_Pre_2011.pdf\" target=\"_blank\" rel=\"noopener\">Part a-1<\/a> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/541a_2012-2024.pdf\" target=\"_blank\" rel=\"noopener\">Part a-2<\/a> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/StatB_Pre_2011.pdf\" target=\"_blank\" rel=\"noopener\">Part b-1 <\/a>\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/11\/541b_2012-2024.pdf\" target=\"_blank\" rel=\"noopener\">Part b-2<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":385,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-2301","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - 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