MAST90105

Jun 8, 2026

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我拿到了一份我在上的课的考前宝典！你帮我拆解一下这节课，告诉我复习重点，我快final了越详细越好，公式，定义，核心考点我都需要，救救孩子

一、先看清这门课到底怎么考
课程结构
- 整门课分成两半：
  - Weeks 1–7：Probability half（概率部分）
  - Weeks 8–12：Inference half（统计推断部分）[2]Source: asksia-bible-mast90105-bilingual.pdfFind a pivotal quantity, invert it; read quantiles off the table "Test Ho VS H," Neyman-Pearson / LRT; or a standard t / x2 / z statistic "Given a prior . . . " Posterior « prior x likelihood; conjugacy; Bayes estimate under the loss ✓ The one habit that wins these papers 拿下这两份试卷的那一个习惯 For every prompt, name the method first, then run its recipe. "Distribution of a function" - CDF / MGF method; "estimate a parameter" - MoM / MLE; "smallest possible variance" - Fisher info - CRLB; "interval for 0" - a pivot; "is the effect real" - a test statistic against a table quantile. The decoder in Ch 14 lists every cue and the recipe it triggers. 对每个题面,先给方法命名,再跑它的recipe。“一个函数的分布”→CDF/ MGF方法;“估计一个参数”→ MoM / MLE;“最小可能variance"→ Fisher info → CRLB;“0的区间”→一个pivot;“效应是否为真”→一个检验统计量对阵一个表上分位数。Ch 14的解码器列出了每个cue及它触发的recipe。 ★ The single highest-value move 单一性价比最高的一招 Spend your A4 sheet on what the provided table cannot give you: the MLE recipe, the CRLB regularity conditions (and when they fail), the Z-t-+x2-+F map, the conjugate-prior table, and the pivot library. The PMF/PDF, MGF, mean and variance for every named distribution are already on the last page - copying them wastes the sheet. Recipes win marks; the table just supplies the constants. 把你的A4花在所提供的表格给不了你的东西上:MLE recipe、CRLB regularity条件(以及它何时失效)、 Z→t→×2→F地图、conjugate-prior表,以及pivot库。每个具名分布的PMF/PDF、MGF、mean与variance 已经在最后一页 -- 抄它们是浪费这张纸。Recipe赢得分数;表格只供给常数。 AskSia Library · MAST90105 · 双语 Bilingual CONTENTS - CONTENTS The whole course, in one ordered book 整门课,凝成一本有序的书 Twelve teaching weeks - one recipe toolkit 十二个教学周→一套recipe工具箱 TL;DR. The book follows the course's two-half arc - the probability half (W1-7: foundations & Bayes, discrete RVs & MGFs, the distribution families, bivariate & correlation, transformations & sampling distributions), which is the mid-semester scope, then the inference half (W8-12: point estimation, estimator properties & the CRLB, Bayesian estimation & regression, interval estimation, hypothesis testing, distribution-free & categorical), which carries the final - before turning it into marks with the glossary, practice bank and exam decoder. TL;DR. 本书沿着课程的两半弧线走 -- probability上半(W1-7:基础与Bayes、discrete RV与MGF、各分布族、bivariate与 correlation、transformation与sampling distribution),也就是期中范围,然后是inference下半(W8-12:点估计、estimator性质与CRLB、Bayesian estimation与regression、区间估计、hypothesis testing、distribution-free与categorical),承载期末 -- 再用glossary、练习库与考试解码器把它变成分数。 Ch Topic Core methods Probability half . Weeks 1-7 (the mid-semester scope) 1 Probability foundations & Bayes sets / counting . conditional probability . total probability & Bayes' theorem . independence → 2 Discrete random variables & MGFs PMF / CDF . expectation, variance · moment generating functions & moments . → independence 3 Discrete & continuous families [10]Source: asksia-bible-mast90105-bilingual.pdfWhole bonus sem Class & Ed participation Examinable spine = the 12-week Hogg-Tanis sequence: probability foundations & Bayes . discrete RVs & MGFs . the discrete families . continuous families . uniform / normal / CLT . bivariate & correlation . transformations & sampling distributions - then estimation . estimator properties & the CRLB . confidence intervals · hypothesis testing · distribution-free & goodness-of-fit. The R lab test draws on the same theory, computed in R. 可考的主干 = 12周的Hogg-Tanis序列:probability基础与 Bayes · discrete RV与MGF · discrete分布族 · continuous分布族 · uniform / normal / CLT · bivariate 5 correlation . transformation5 sampling distribution -然后是estimation · estimator性质与CRLB · confidence interval · hypothesis testing · distribution- free与goodness-of-fit。R lab test取材于同一套理论,只是在R里计算。 FIG 0. 1 log-likelihood |(0) I'(e) = O (score) L(8). C MLE = max order stat θ {""(e) < 0 - maximum -¿** (e) = observed info 0 (MLE) θ The inference half in one picture: a log-likelihood {(0) that peaks at the MLE 0, where the score {' (o) = 0 and the curvature {"(0) < 0 confirms a maximum (and gives the observed information). The gold inset is the boundary case - Unif(0,0), whose likelihood is maximised at the largest order statistic, not by setting a derivative to zero. The whole MLE recipe lives in this one curve; learn to draw it from memory. 推断部分一图概览:log-likelihood e(8) 在 MLE 0^ 处取得峰值,此处 score l'(0)= 0,且曲率 Q"(日)< 0 确认这是极大值(并给出 observed information)。金色嵌图是边界情形 -- Unif(0,0), 其 likelihood 在最大 order statistic 处取得极大,而非通过令导数为零。整套MLE 流程都浓缩在这一条曲线里;要练到能凭记忆画出来。 AskSia Library · MAST90105 · 双语 Bilingual boundary case Unif(0, 0) What the papers are really testing 这两份试卷到底在考什么 The cue you get The recipe it rewards "Find the distribution of Y = g(X)" CDF method or MGF method - name the resulting family "Estimate 0" MoM (match moments) and/or MLE (L - In L - score = 0) "Best possible variance?" Fisher information - CRLB - and check the regularity conditions "Construct a CI"
考试结构
- mid-sem：35%，主要考 Weeks 1–7 概率部分[1]Source: asksia-bible-mast90105-bilingual.pdfX2 F θ - THE COMPLETE EXAM BIBLE Methods of Mathematical Statistics 数理统计方法 THE TABLE IS HANDED TO YOU - SO MEMORISE THE RECIPES. WRITE LLNL- SCORE = 0; PICK THE PIVOT; NAME THE DISTRIBUTION. 带一张 A4 进考场 -- 记配方,不抄分布表。 MAST90105 . THE UNIVERSITY OF MELBOURNE 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行一考试要点与术语保留英文原词 Two written exams carry 70%: a mid-semester on the probability half (Weeks 1-7, 35%) and a cumulative final weighted to the inference half (Weeks 8-12, 35%). Each is 3 hours, and into each you may carry one A4 double-sided handwritten or printed sheet plus a non- programmable Casio FX-82. Crucially, a table of every distribution's PMF/PDF, MGF, mean and variance is printed on the last page of the paper. So your sheet should never copy that table - it should carry the derivation recipes and decision logic the table cannot give you. A separate 10% R lab test is open-book. Independent study companion. Not affiliated with or endorsed by the University of Melbourne. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Recipes you derive, not numbers you copy 你推导出来的recipe,而不是抄来的数字 The table is provided - this book is everything the table is not 表格已发给你 -- 这本书全是表格里没有的东西 TL;DR. This is not a re-print of Hogg-Tanis or a dump of the distribution table - that table is handed to you in the exam. It is a self-contained bank of the derivation recipes, decision trees and worked exam- type cases the two written papers actually reward: every estimator, score, information bound, pivot and test statistic typeset and derived, each method drawn as an original schematic where a picture helps, and tied to the exam's one move - name the method, set it up, then read the numbers off the provided table. The same pages serve you three ways across the twelve teaching weeks. TL;DR. 这不是Hogg-Tanis的翻印,也不是distribution table的堆砌 -- 那张表考试时会发给你。它是一个自成体系的库,装着两份笔试真正给分的东西:推导recipe、决策树与考试题型的例题:每一个estimator、score、information bound、pivot与检验统计量都经过排版与推导,凡是图能帮忙的地方,每种方法都画成原创示意图,并紧扣考试的那一招 -- 给方法命名、把它建立起来,再从所提供的表格上读出数字。同样这些页面,在这十二个教学周里能以三种方式服务于你。 A 1 . LEARN 1 ·LEARN(学) You haven't done the week's lecture yet. Read a chapter top to bottom. Each method opens with a plain-English definition, lands a typeset derivation or a decision table, then a worked example with our numbers that shows the full recipe - L -> In L - score = 0, or pick the pivot, or posterior « prior x likelihood. Meet the MLE, the CRLB, the pivot and Neyman-Pearson here cold. 你这周的lecture还没上。把一章从头读到尾。每种方法以一个通俗的定义开场,落到一段排版的推导或一张决策表,再接一个用我们数字的例题,展示完整的recipe -L → In L → score = 0,或挑 pivot, ¿¿ posterior % prior x likelihood。在这里冷启动地认识 MLE, CRLB, pivot5 Neyman-Pearson. B 2 . REVISE 2 · REVISE(复习) You've done the week. Use the grids and the chapter-end recall checklists to self-test: can you write the MLE recipe, state the regularity conditions for the CRLB, list the Z-t-+x2-+F relationships, recall the conjugate priors? The checklists are written to be lifted almost verbatim onto your one A4 double-sided sheet. 你这周上完了。用各网格与章末的回想清单来自测:你能写出MLE recipe吗?能陈述CRLB的 regularity条件吗?能列出 Z→t→×2→F关系吗?能回想起 conjugate prior吗?这些清单写出来就是为了几乎逐字搬上你那张A4双面纸。[3]Source: asksia-bible-mast90105-bilingual.pdfC 3 . APPLY 3 · APPLY(应用) You're building your A4 sheet or sitting a paper. Run the name- the-method decoder (Ch 14) on every prompt: read the cue ++ name the method (transformation? MLE? pivot? test?) - set up the maths - read constants off the provided table and the Casio FX-82. Your edge is recipe fluency, not memorised formulae. 你在搭A4或正坐考。对每个题面跑name-the-method解码器 (Ch 14):读cue → 给方法命名 (transformation?MLE?pivot? test?)→ 把数学搭起来 → 从所提供的表格与Casio FX-82上读出常数。你的优势是recipe的熟练,而非背下的公式。 AskSia Library · MAST90105 · 双语 Bilingual ! Read this first: the assessment shape, and the bring-in rule 先读这个:考核形态,以及带入规则 MAST90105 is assessed by four pieces plus a small bonus: 4 written assignments (20%, 5% each), the mid- semester exam (35%, the probability half, Weeks 1-7), a 10% computer / R lab test (open-book, laptop + R, no communication), the final exam (35%, cumulative but weighted to the inference half, Weeks 8-12), and up to 5% engagement bonus. Both written exams are on-campus, invigilated, 3 hours (15 min reading + 3 hr writing). Into each you may carry one A4 double-sided handwritten or printed sheet and a non-programmable Casio FX-82, and a distribution table is appended to the paper. The R lab test is the only open-book task. So your A4 sheet should carry recipes, decision rules and the formulae the table omits, never the table itself. Always confirm current weights, dates and exam conditions on your own LMS, as details shift between cohorts. MAST90105由四项加一个小bonus考核:4份书面assignment(20%,各5%)、期中考(35%,probability上半,Weeks 1- 7)、一个10%的computer / R lab test(开卷,笔记本+R,不得交流)、期末考(35%,累积但偏重inference下半,Weeks 8- 12),以及最高5%的参与bonus。两场笔试都是校内、监考、3小时(15分钟阅读+3小时书写)。每场你可带入一张A4双面手写或打印纸与一台不可编程的Casio FX-82,而且试卷附有一张distribution table。R lab test是唯一的开卷任务。所以你的A4应装上recipe、决策规则与表格略去的公式,绝不放表格本身。务必在你自己的LMS上确认当前权重、日期与考试条件,因为细节会随届次变动。 i How this book was built - the two-layer rule 这本书是怎么搭起来的 -- 两层规则 The theory canon here is standard, widely-published mathematical statistics - the results in Hogg, Tanis & Zimmerman (9e), and the same classical canon in Wackerly and Casella-Berger: the MLE and method-of-moments recipes, the CDF- and MGF-methods for transformations, Fisher information and the Cramer-Rao lower bound, pivotal-quantity confidence intervals, the Neyman-Pearson lemma and likelihood-ratio tests, Bayesian posterior reasoning with conjugacy, and the Z-t-x2-F sampling-distribution relationships. These are non-copyrightable canon, stated and derived plainly. The course's own exam, assignment and practice questions are paraphrased and re-authored with AskSia-invented stems, parameters and numbers - we never reproduce a question verbatim. Book status quoted and honoured (one A4 double-sided sheet, Casio FX-82, table provided). Verify on your LMS. 这里的理论正典是标准、广为出版的mathematical statistics -- Hogg, Tanis & Zimmerman(9e)里的结果,以及 Wackerly与Casella-Berger里同样的经典正典:MLE与method-of-moments recipe、变换的CDF与MGF方法、Fisher information 5 Cramer-Rao lower bound, pivotal-quantity confidence interval, Neyman-Pearson lemma5 likelihood-ratio test、带共轭性的Bayesian posterior推理,以及Z→t→×2→F的sampling-distribution关系。这些是不可受版权保护的正典,平实地陈述与推导。本课自己的考试、assignment与练习题都被改写并以AskSia自拟的题干、参数与数字重新编写 -- 我们绝不逐字复制任何一题。所引用并遵循的考试状态(一张A4双面纸、Casio FX-82、提供表格)。请在你的LMS上核实。 AskSia Library · MAST90105 · 双语 Bilingual THE BLUEPRINT - THE EXAM BLUEPRINT 70% IN TWO PAPERS . MEMORISE RECIPES, NOT THE TABLE Where every mark lives 每一分都落在哪里 Two 3-hour written papers - mid-sem (probability) 35% + final (inference) 35% - each with one A4 sheet, a Casio FX-82, and a provided distribution table 两场3小时笔试 -- 期中(probability)35%+期末(inference)35% -- 每场一张A4纸、一台Casio FX-82,并提供一张distribution table TL;DR. Seventy percent sits in two written exams: a mid-semester on the probability half (Weeks 1-7) and a cumulative final weighted to the inference half (Weeks 8-12), each 35%. Into each you carry one A4 double-sided sheet + a Casio FX-82, and a distribution table is printed on the paper. So the winning skill is not recall - it is setting up the right recipe (transformation, MLE, CRLB, pivot, test) and then reading the constants off the table. Master the recipes and decision trees in this book and you hold the keys to both papers. TL;DR. 七成分数落在两场笔试:一场考probability上半(Weeks 1-7)的期中,一场偏重inference下半(Weeks 8-12)的累积性期末,各占35%。每场你可带入一张A4双面纸+一台Casio FX-82,而且试卷上印有一张distribution table。所以制胜技能不是死记 -- 而是把对的recipe搭起来(transformation、MLE、CRLB、pivot、检验),然后从表格上读出常数。把这本书里的 recipe与决策树吃透,你就握住了两份试卷的钥匙。 35+35% TWO WRITTEN EXAMS 两场笔试 1 A4 DOUBLE-SIDED SHEET A4 双面纸 FX-82 ✓ CASIO (NON-PROGRAMMABLE) Casio (非可编程) TABLE IS PROVIDED 提供表格 AskSia Library · MAST90105 · 双语 Bilingual The assessment pieces
- final：35%，累积性考试，但重点偏向 Weeks 8–12 的 inference half，同时默认你会前面的概率基础[1]Source: asksia-bible-mast90105-bilingual.pdfX2 F θ - THE COMPLETE EXAM BIBLE Methods of Mathematical Statistics 数理统计方法 THE TABLE IS HANDED TO YOU - SO MEMORISE THE RECIPES. WRITE LLNL- SCORE = 0; PICK THE PIVOT; NAME THE DISTRIBUTION. 带一张 A4 进考场 -- 记配方,不抄分布表。 MAST90105 . THE UNIVERSITY OF MELBOURNE 中英双语版 · BILINGUAL EDITION 英文主讲,中文随行一考试要点与术语保留英文原词 Two written exams carry 70%: a mid-semester on the probability half (Weeks 1-7, 35%) and a cumulative final weighted to the inference half (Weeks 8-12, 35%). Each is 3 hours, and into each you may carry one A4 double-sided handwritten or printed sheet plus a non- programmable Casio FX-82. Crucially, a table of every distribution's PMF/PDF, MGF, mean and variance is printed on the last page of the paper. So your sheet should never copy that table - it should carry the derivation recipes and decision logic the table cannot give you. A separate 10% R lab test is open-book. Independent study companion. Not affiliated with or endorsed by the University of Melbourne. Corrections: takedowns@asksia. ai PREFACE - HOW TO USE THIS BOOK Recipes you derive, not numbers you copy 你推导出来的recipe,而不是抄来的数字 The table is provided - this book is everything the table is not 表格已发给你 -- 这本书全是表格里没有的东西 TL;DR. This is not a re-print of Hogg-Tanis or a dump of the distribution table - that table is handed to you in the exam. It is a self-contained bank of the derivation recipes, decision trees and worked exam- type cases the two written papers actually reward: every estimator, score, information bound, pivot and test statistic typeset and derived, each method drawn as an original schematic where a picture helps, and tied to the exam's one move - name the method, set it up, then read the numbers off the provided table. The same pages serve you three ways across the twelve teaching weeks. TL;DR. 这不是Hogg-Tanis的翻印,也不是distribution table的堆砌 -- 那张表考试时会发给你。它是一个自成体系的库,装着两份笔试真正给分的东西:推导recipe、决策树与考试题型的例题:每一个estimator、score、information bound、pivot与检验统计量都经过排版与推导,凡是图能帮忙的地方,每种方法都画成原创示意图,并紧扣考试的那一招 -- 给方法命名、把它建立起来,再从所提供的表格上读出数字。同样这些页面,在这十二个教学周里能以三种方式服务于你。 A 1 . LEARN 1 ·LEARN(学) You haven't done the week's lecture yet. Read a chapter top to bottom. Each method opens with a plain-English definition, lands a typeset derivation or a decision table, then a worked example with our numbers that shows the full recipe - L -> In L - score = 0, or pick the pivot, or posterior « prior x likelihood. Meet the MLE, the CRLB, the pivot and Neyman-Pearson here cold. 你这周的lecture还没上。把一章从头读到尾。每种方法以一个通俗的定义开场,落到一段排版的推导或一张决策表,再接一个用我们数字的例题,展示完整的recipe -L → In L → score = 0,或挑 pivot, ¿¿ posterior % prior x likelihood。在这里冷启动地认识 MLE, CRLB, pivot5 Neyman-Pearson. B 2 . REVISE 2 · REVISE(复习) You've done the week. Use the grids and the chapter-end recall checklists to self-test: can you write the MLE recipe, state the regularity conditions for the CRLB, list the Z-t-+x2-+F relationships, recall the conjugate priors? The checklists are written to be lifted almost verbatim onto your one A4 double-sided sheet. 你这周上完了。用各网格与章末的回想清单来自测:你能写出MLE recipe吗?能陈述CRLB的 regularity条件吗?能列出 Z→t→×2→F关系吗?能回想起 conjugate prior吗?这些清单写出来就是为了几乎逐字搬上你那张A4双面纸。[2]Source: asksia-bible-mast90105-bilingual.pdfFind a pivotal quantity, invert it; read quantiles off the table "Test Ho VS H," Neyman-Pearson / LRT; or a standard t / x2 / z statistic "Given a prior . . . " Posterior « prior x likelihood; conjugacy; Bayes estimate under the loss ✓ The one habit that wins these papers 拿下这两份试卷的那一个习惯 For every prompt, name the method first, then run its recipe. "Distribution of a function" - CDF / MGF method; "estimate a parameter" - MoM / MLE; "smallest possible variance" - Fisher info - CRLB; "interval for 0" - a pivot; "is the effect real" - a test statistic against a table quantile. The decoder in Ch 14 lists every cue and the recipe it triggers. 对每个题面,先给方法命名,再跑它的recipe。“一个函数的分布”→CDF/ MGF方法;“估计一个参数”→ MoM / MLE;“最小可能variance"→ Fisher info → CRLB;“0的区间”→一个pivot;“效应是否为真”→一个检验统计量对阵一个表上分位数。Ch 14的解码器列出了每个cue及它触发的recipe。 ★ The single highest-value move 单一性价比最高的一招 Spend your A4 sheet on what the provided table cannot give you: the MLE recipe, the CRLB regularity conditions (and when they fail), the Z-t-+x2-+F map, the conjugate-prior table, and the pivot library. The PMF/PDF, MGF, mean and variance for every named distribution are already on the last page - copying them wastes the sheet. Recipes win marks; the table just supplies the constants. 把你的A4花在所提供的表格给不了你的东西上:MLE recipe、CRLB regularity条件(以及它何时失效)、 Z→t→×2→F地图、conjugate-prior表,以及pivot库。每个具名分布的PMF/PDF、MGF、mean与variance 已经在最后一页 -- 抄它们是浪费这张纸。Recipe赢得分数;表格只供给常数。 AskSia Library · MAST90105 · 双语 Bilingual CONTENTS - CONTENTS The whole course, in one ordered book 整门课,凝成一本有序的书 Twelve teaching weeks - one recipe toolkit 十二个教学周→一套recipe工具箱 TL;DR. The book follows the course's two-half arc - the probability half (W1-7: foundations & Bayes, discrete RVs & MGFs, the distribution families, bivariate & correlation, transformations & sampling distributions), which is the mid-semester scope, then the inference half (W8-12: point estimation, estimator properties & the CRLB, Bayesian estimation & regression, interval estimation, hypothesis testing, distribution-free & categorical), which carries the final - before turning it into marks with the glossary, practice bank and exam decoder. TL;DR. 本书沿着课程的两半弧线走 -- probability上半(W1-7:基础与Bayes、discrete RV与MGF、各分布族、bivariate与 correlation、transformation与sampling distribution),也就是期中范围,然后是inference下半(W8-12:点估计、estimator性质与CRLB、Bayesian estimation与regression、区间估计、hypothesis testing、distribution-free与categorical),承载期末 -- 再用glossary、练习库与考试解码器把它变成分数。 Ch Topic Core methods Probability half . Weeks 1-7 (the mid-semester scope) 1 Probability foundations & Bayes sets / counting . conditional probability . total probability & Bayes' theorem . independence → 2 Discrete random variables & MGFs PMF / CDF . expectation, variance · moment generating functions & moments . → independence 3 Discrete & continuous families [3]Source: asksia-bible-mast90105-bilingual.pdfC 3 . APPLY 3 · APPLY(应用) You're building your A4 sheet or sitting a paper. Run the name- the-method decoder (Ch 14) on every prompt: read the cue ++ name the method (transformation? MLE? pivot? test?) - set up the maths - read constants off the provided table and the Casio FX-82. Your edge is recipe fluency, not memorised formulae. 你在搭A4或正坐考。对每个题面跑name-the-method解码器 (Ch 14):读cue → 给方法命名 (transformation?MLE?pivot? test?)→ 把数学搭起来 → 从所提供的表格与Casio FX-82上读出常数。你的优势是recipe的熟练,而非背下的公式。 AskSia Library · MAST90105 · 双语 Bilingual ! Read this first: the assessment shape, and the bring-in rule 先读这个:考核形态,以及带入规则 MAST90105 is assessed by four pieces plus a small bonus: 4 written assignments (20%, 5% each), the mid- semester exam (35%, the probability half, Weeks 1-7), a 10% computer / R lab test (open-book, laptop + R, no communication), the final exam (35%, cumulative but weighted to the inference half, Weeks 8-12), and up to 5% engagement bonus. Both written exams are on-campus, invigilated, 3 hours (15 min reading + 3 hr writing). Into each you may carry one A4 double-sided handwritten or printed sheet and a non-programmable Casio FX-82, and a distribution table is appended to the paper. The R lab test is the only open-book task. So your A4 sheet should carry recipes, decision rules and the formulae the table omits, never the table itself. Always confirm current weights, dates and exam conditions on your own LMS, as details shift between cohorts. MAST90105由四项加一个小bonus考核:4份书面assignment(20%,各5%)、期中考(35%,probability上半,Weeks 1- 7)、一个10%的computer / R lab test(开卷,笔记本+R,不得交流)、期末考(35%,累积但偏重inference下半,Weeks 8- 12),以及最高5%的参与bonus。两场笔试都是校内、监考、3小时(15分钟阅读+3小时书写)。每场你可带入一张A4双面手写或打印纸与一台不可编程的Casio FX-82,而且试卷附有一张distribution table。R lab test是唯一的开卷任务。所以你的A4应装上recipe、决策规则与表格略去的公式,绝不放表格本身。务必在你自己的LMS上确认当前权重、日期与考试条件,因为细节会随届次变动。 i How this book was built - the two-layer rule 这本书是怎么搭起来的 -- 两层规则 The theory canon here is standard, widely-published mathematical statistics - the results in Hogg, Tanis & Zimmerman (9e), and the same classical canon in Wackerly and Casella-Berger: the MLE and method-of-moments recipes, the CDF- and MGF-methods for transformations, Fisher information and the Cramer-Rao lower bound, pivotal-quantity confidence intervals, the Neyman-Pearson lemma and likelihood-ratio tests, Bayesian posterior reasoning with conjugacy, and the Z-t-x2-F sampling-distribution relationships. These are non-copyrightable canon, stated and derived plainly. The course's own exam, assignment and practice questions are paraphrased and re-authored with AskSia-invented stems, parameters and numbers - we never reproduce a question verbatim. Book status quoted and honoured (one A4 double-sided sheet, Casio FX-82, table provided). Verify on your LMS. 这里的理论正典是标准、广为出版的mathematical statistics -- Hogg, Tanis & Zimmerman(9e)里的结果,以及 Wackerly与Casella-Berger里同样的经典正典:MLE与method-of-moments recipe、变换的CDF与MGF方法、Fisher information 5 Cramer-Rao lower bound, pivotal-quantity confidence interval, Neyman-Pearson lemma5 likelihood-ratio test、带共轭性的Bayesian posterior推理,以及Z→t→×2→F的sampling-distribution关系。这些是不可受版权保护的正典,平实地陈述与推导。本课自己的考试、assignment与练习题都被改写并以AskSia自拟的题干、参数与数字重新编写 -- 我们绝不逐字复制任何一题。所引用并遵循的考试状态(一张A4双面纸、Casio FX-82、提供表格)。请在你的LMS上核实。 AskSia Library · MAST90105 · 双语 Bilingual THE BLUEPRINT - THE EXAM BLUEPRINT 70% IN TWO PAPERS . MEMORISE RECIPES, NOT THE TABLE Where every mark lives 每一分都落在哪里 Two 3-hour written papers - mid-sem (probability) 35% + final (inference) 35% - each with one A4 sheet, a Casio FX-82, and a provided distribution table 两场3小时笔试 -- 期中(probability)35%+期末(inference)35% -- 每场一张A4纸、一台Casio FX-82,并提供一张distribution table TL;DR. Seventy percent sits in two written exams: a mid-semester on the probability half (Weeks 1-7) and a cumulative final weighted to the inference half (Weeks 8-12), each 35%. Into each you carry one A4 double-sided sheet + a Casio FX-82, and a distribution table is printed on the paper. So the winning skill is not recall - it is setting up the right recipe (transformation, MLE, CRLB, pivot, test) and then reading the constants off the table. Master the recipes and decision trees in this book and you hold the keys to both papers. TL;DR. 七成分数落在两场笔试:一场考probability上半(Weeks 1-7)的期中,一场偏重inference下半(Weeks 8-12)的累积性期末,各占35%。每场你可带入一张A4双面纸+一台Casio FX-82,而且试卷上印有一张distribution table。所以制胜技能不是死记 -- 而是把对的recipe搭起来(transformation、MLE、CRLB、pivot、检验),然后从表格上读出常数。把这本书里的 recipe与决策树吃透,你就握住了两份试卷的钥匙。 35+35% TWO WRITTEN EXAMS 两场笔试 1 A4 DOUBLE-SIDED SHEET A4 双面纸 FX-82 ✓ CASIO (NON-PROGRAMMABLE) Casio (非可编程) TABLE IS PROVIDED 提供表格 AskSia Library · MAST90105 · 双语 Bilingual The assessment pieces [16]Source: asksia-cheatsheet-mast90105.pdfMAST90105 Methods of Mathematical Statistics UNIVERSITY OF MELBOURNE . SCHOOL OF MATHEMATICS & STATISTICS BRING-IN A4 . DISTRIBUTION TABLE PROVIDED . MID-SEM + FINAL Sem 1 2026 . SIDE 1 OF 2 Probability half · Weeks 1-7 (mid-sem) SIDE 1/2 map 0 . How to Use This A4 READ FIRST * The mid-sem & final are written, bring-in: one A4 double-sided sheet + a non-programmable Casio FX- 82, and a distribution table is provided on the last page (every PMF/PDF, MGF, mean, variance). So do NOT copy the table . This sheet holds what the table does not give: the derivation recipes, the decision logic, and the off-table formulas (sampling- distribution links, score equations, CRLB, pivots). Split: Side 1 = probability (mid-sem, W1-7); Side 2 = inference (final, W8-12, cumulative). Read the numbers off the table, run the recipe from here. The final is cumulative - it assumes all of W1-7 but weights the inference half. SIA > Marker reflex: name the distribution & state the method before you compute - "by the CDF method . . . ", "MLE: score=0 then {"<0 . . . ". The recipe earns the marks; the table earns nothing. Carry the table's notation onto your sheet so you can cross-reference fast under time pressure. 1 . Probability & Bayes Axioms: Pr(A)≥0, Pr(S)=1, countable additivity. Pr(Ai)=1-Pr(A); Pr(AuB)=Pr(A)+Pr(B)-Pr(AnB). COUNTING perms n!/(n-r) ! . combos C(n,r)=n!/[r! (n-r) !] Conditional & total prob: Pr(A|B)=Pr (AnB) /Pr (B) Pr (A)=ΣK Pr (A/BK)Pr (BK) (partition) * BAYES' THEOREM Pr(B; |A) = Pr (A|B;)Pr(B;) / EK Pr(A|BK)Pr (Bk) Recipe: @ find the "which-source" partition (which coin/die/box; priors often all equal) @ likelihood of the evidence under each source @ plug in & renormalise by the denominator . Independence: ALB = Pr(AnB)=Pr(A)Pr(B). Pairwise # mutual. 1b . Worked . Bayes source OUR NUMBERS Three coins, Pr(tail)=0. 5, 0. 3, 0. 7; pick one (prior 1/3 each), toss till first tail - tail on toss 3 (HHT). |Lk=(1-PK)2PK - 0. 125, 0. 147, 0. 063 Posterior fair coin = 0. 125/(0. 125+0. 147+0. 063) = 0. 373. Trap: quoting Pr(E|C) as the answer (forgot to renormalise); reading "till first tail" as binomial not geometric. 1c . Counting reflexes URN PROBLEMS For without-replacement PMFs, count favourable elementary outcomes + total: · Order matters => permutations n!/(n-r)! Order ignored => combinations C(n,r) . Both types of object => product of two combinations (hypergeometric shape) INCLUSION-EXCLUSION (3 SETS) | AUBUC | = Σ |ΑΙ - Σ | ΑΛΒΙ + | AnBnC| Trap: mixing ordered with unordered counts in the same ratio; forgetting C(n,r)=C(n,n-r). 1d . Total prob vs Bayes SPOT WHICH Two readings of one stem: "what is Pr(evidence)?" => total probability (the Bayes denominator); "which source?" = Bayes (the full ratio). 2 . Discrete RV . Moments . MGF W2
final 的出题核心
- 题目通常不是问你“背了什么”，而是问你：
  - 这个题属于哪种方法？
  - 你会不会把这个方法的标准 recipe 写出来？
  - 你会不会用给定表格读分位数/常数？
- 材料里反复强调一条习惯：
  - 先写方法名，再开始算。
  - 比如：

1）Likelihood / MLE

2）Method of Moments (MoM)

3）Bias

4）Variance

5）MSE

6）Fisher Information

定义公式
- 材料给出的核心写法： $$ I(\theta)=-E!\left[\frac{\partial^2}{\partial \theta^2}\log f(X;\theta)\right] $$
- 样本量为 $n$ 时，CRLB 用 $nI(\theta)$[5]Source: asksia-bible-mast90105-bilingual.pdf"Estimate the parameter (" Method of Moments, then MLE (1) MOM: set E(X) =- solve ÔMM; (2) MLE: l'(0) = 0 - ÔML ; (3) check !" < 0 (or read a boundary / order statistic). "Is this the best / most efficient estimator? Find its variance" Bias, variance & the Cramér-Rao bound (1) bias: E() - 0; (2) I(0) = - E[0} In f], CRLB = 1/(nl); (3) hits the bound & unbiased => efficient (check regularity!). "Construct a confidence interval for 0" Pivotal quantity - invert (1) find a pivot V (data, 0) with a 0-free law; (2) Pr(a ≤ V ≤ b) = 1 - a; (3) invert algebraically to bracket 0. "Test Ho VS Ha / find the most powerful test" Neyman-Pearson / LRT, or a standard t / x2 / z (1) simple-vs-simple: reject when L(00)/L(01) < k; (2) set c from a (or a = B); (3) else pick the standard statistic (see D. 2) & get the p-value. "A prior distribution is given" Bayesian posterior & priorxlikelihood (1) TT(0|x)&T(0)L(0)- spot conjugacy; (2) Bayes estimate = posterior mean (squared loss) or median (absolute loss); (3) credible interval from posterior quantiles. "Do these categorical counts fit / are the factors independent?" Chi-square goodness- of-fit / contingency (1) expected E; (GOF) or Eij = RICi; (2) x2 = _ (0 - E)2/ E; (3) df = k -1 -mor (r-1)(c-1). ! Three cues that look alike but split the method 三个看着相像、却分流方法的cue (1) "Estimate" vs "best estimator" - the first wants MoM/MLE, the second wants bias / variance / CRLB. (2) "Confidence" vs "credible" interval - pivot-invert is frequentist; a prior in the stem switches you to the Bayesian posterior quantiles. (3) Test cue - "most powerful" / "simple vs simple" = Neyman-Pearson; an unspecified mean/variance/proportion = the standard t / x2 / z in D. 2. (1) “Estimate(估计)”对“best estimator(最佳估计量)”––前者要 MoM/MLE,后者要 bias / variance / CRLB。(2) "Confidence(置信)”对“credible(可信)”区间 -- pivot 反演是频率派的;题干里出现 prior (先验) 就把你切换到贝叶斯后验分位数。(3)检验线索 -- “most powerful(最强势)”/ “simple vs simple"⇒ Neyman- Pearson;一个未指明的均值/方差/比例⇒ D. 2 里的标准 t/ x2 / z。 AskSia Library · MAST90105 · 双语 Bilingual EXAM MORNING . THE DECODER EXAM MORNING . THE DECODER WHAT TO WRITE . WHAT IS ALREADY PROVIDED The bring-in A4 plan & the recipes to recall 带入的A4方案与要回想的recipe TL;DR. The exam prints the distribution table (every PMF/PDF, MGF, mean and variance for the standard families), so your one A4 double-sided sheet must carry the three things it does not. the derivation recipes, the decision logic, and the handful of off-table formulas (the Z-t-x2-F links, CRLB, the standard test statistics, the posterior updates). Below is the layout, the mid-sem/final split, and the family map - the last picture to glance at before you walk in.[6]Source: asksia-bible-mast90105-bilingual.pdf把这个WRITE下来(表格不给) · Recipes: Bayes; CDF/MGF transformation; MOM & the 3-step MLE (L > e > l' = 0 > l" < 0); boundary/order-statistic MLE. 配方:Bayes; CDF/MGF 变换;MOM & 三步 MLE;边界/ order-statistic MLE。 - · Off-table formulas: I(0) = - E[02 In f], CRLB = 1/(nI); the four sampling-family links (Z, x2, t, F); the D. 2 test statistics; posterior & priorxlikelihood + the Beta-Bernoulli/geometric conjugate update. 表外公式:,CRLB;四条抽样分布族联系(Z、 ×2、t、F); D. 2 检验统计量;posterior α priorxlikelihood + Beta-Bernoulli/geometric # 轭更新。 · Decision logic: the D. 1 cue grid + o known++Z / estimated-t / variance++x2 selector + regularity- fails-for-Unif(0,0) flag. 决策逻辑:D. 1 线索网格 +α 已知→Z/ 估计→t / variance→×2 选择器+ “均匀分布 regularity 失效”标记。 ● Recipe(套路):Bayes;CDF/MGF 变换;MOM 与三步 MLE(L→l→l'=0→l"<0);边界 / order-statistic MLE. ● 表外公式:I(0) =- E[8% In f], CRLB =1/(nI);四条抽样族链接(Z,x2,t,F);D. 2 检验统计量;后验 α priorxlikelihood + Beta-Bernoulli/geometric # 轭更新。 ,决策逻辑: D. 1 线索网格 + o已知→Z/ 被估计→t / 方差→×2 选择器 + regularity-对-Unif(0,0)-失效的标记。 ! DO NOT waste A4 space on this 别在这上面浪费A4空间 · The PMF/PDF, MGF, mean and variance of Bernoulli, Binomial, Geometric, Neg-Binomial, Hypergeometric, Poisson, discrete Uniform, Beta, x2, Exponential, Gamma, continuous Uniform and Normal - all printed on the last page. Bernoulli, Binomial, Geometric, Neg- Binomial、Hypergeometric、Poisson、离散 Uniform、Beta、x2、Exponential、Gamma、连续 Uniform 与 Normal 的 PMF/PDF、MGF、 mean 与 variance -- 全部印在最后一页。 FX-82 或配方本身就能复现的那些冗长积分表。 · R quantile values (qnorm, qbeta, Pt, . . . ) - the paper supplies the ones you need. I R 的分位数值 -- 试卷会提供你需要的那些。 · Bernoulli, Binomial, Geometric, Neg- Binomial、Hypergeometric、Poisson、离散 Uniform、Beta、x2、Exponential、Gamma、连续 Uniform 与 Normal 的 PMF/PDF、MGF、均值与方差 -- 全部印在最后一页。 ● FX-82 或 recipe 会重现的长篇积分表。 R 分位数值(qno qbeta, Pt; . . ) -- 卷子会供给你所需的那些。 D. 4 The two-paper split - where each method is examined D. 4两卷划分 -- 每种方法在哪里被考 Paper Scope Methods to drill Mid-sem (35%, 3 h) Probability half - Weeks 1-7 Bayes; PMFs by enumeration; Poisson superposition; MGF-moments; piecewise- PDF constant/mean/median; bivariate p & the "p = 0 / indep" trap; transformations. Final (35%, 3 h, Jun 9-26) Inference half - Weeks 8-12 (assumes 1-7)[13]Source: asksia-bible-mast90105-bilingual.pdfMoM & MLE; bias/variance; Fisher info & CRLB (+regularity); Bayesian posterior + credible interval; pivot Cls; Neyman-Pearson; the t / x2 / z standard tests; GOF / contingency. D. 5 The last glance: the sampling-distribution family D. 5最后一瞥:sampling-distribution分布族 AskSia Library · MAST90105 · 双语 Bilingual FIG D. 1 X2k chi-square, k df sum of k Z2 X k = 2 21 tk Student t, k df tk = Z / V(X2k / k) Z N(0, 1) F = (X2m/m) + (xªn/n) Fmin ratio, (m, n) df as df - c : tk - Z and X2 k/k + 1 (F + const) table gives each density The one map both papers lean on: a standard normal Z at the centre; square and sum k of them > X2k; Z/Vx}/k- tk; a ratio of two scaled X2 > Fm,n. The provided table gives each density; these relationships are the off- table facts that drive every CI and test. 两份试卷都依赖的那张图:中心是 standard normal Z;把它们平方求和→×2k;→ tk;两个缩放x2 之比→ Fm,no 所提供的表格给出各density;这些关系才是表上没有、却驱动每个 CI 与检验的事实。 AskSia Library · MAST90105 · 双语 Bilingual ★ Recipes to recall - the whole-course checklist 要回想的recipe -- 全课程清单 · Bayes: priorxlikelihood, renormalise by total probability. Bayes: priorxlikelihood,按 total probability 重新归一化。 · MLE: {' = 0 then !" < 0; watch for a boundary / order-statistic MLE. MOM: E(X) = x. MLE:先后;当心边界 / order-statistic MLE。MOM: 。 · CRLB: Var(0) ≥ 1/(nI(0)); regularity fails for Unif(0,0). CRLB :; 对均匀分布regularity 失效。 · Bayesian: conjugate posterior - mean (squared loss) / median (absolute loss) + credible interval. Bayesian: 共轭 posterior → mean (平方损失)/ median(绝对损失)+ credible interval。 · CI: find a pivot, Pr(a ≤ V ≤ b) = 1 - a, invert. CI:找一个 pivot,,反解。 · Tests: N-P likelihood ratio (set c from a or a = B); standard t / x2 / z (D. 2); p-value vs a. Tests: N-P likelihood ratio (由 α或设c);标准 t/ x2 / z (D. 2); p-value 对照 α。

7）Cramér–Rao Lower Bound (CRLB)

8）Pivotal Quantity

9）Confidence Interval (CI)

构造逻辑

10）Posterior

11）Bayes estimate

12）p-value / $\alpha$ / power

五、final 最高频解题链条

A. 参数估计题：MoM + MLE

B. 估计量优劣题：bias + variance + MSE + CRLB

C. 区间估计题：pivot → bracket → invert

D. 假设检验题：选对统计量比计算更重要

核心思想：
- 先判断这是哪类检验，再写统计量，再看参考分布，再和临界值/p-value 比较。[4]Source: asksia-bible-mast90105-bilingual.pdf拟合优度 / 列联表 x2 = >(O; - E;)2/E; ~ x2-1-mi tests fit / factor independence. Distribution-free (nonparametric) 非参数方法 Inference using ranks / order statistics (sign, Wilcoxon); no distributional assumption. ★ What the exam asks here 这里考试问什么 The bring-in written exams (mid-sem 35% = probability half W1-7; final 35% = inference half W8-12; each allows one A4 double-sided sheet + a Casio FX-82, with the distribution table provided) reward precise vocabulary. Recurring mark-earners: stating the MLE recipe (l' = 0, check {" < 0), the CRLB with its regularity caveat, naming the pivot before inverting a CI, and not confusing @ / p-value / power. Memorise the one-line phrasing - the marks are in the wording, not the table. 可带笔记的笔试(mid-sem 35%=概率部分 W1-7;final 35% = 推断部分 W8-12;每场都允许一张 A4 双面纸 + 一台 Casio FX-82,且提供 distribution table)奖励精准的术语。反复出现的得分点:陈述 MLE recipe(,核对)、带 regularity 警告的 CRLB、在反演一个 CI 之前命名 pivot,以及不要混淆 α / p-value / power。把这一行话术背下来 -- 分数在措辞里,不在表里。 AskSia Library · MAST90105 · 双语 Bilingual CH13 . PRACTICE BANK CH13 . PRACTICE BANK HOGG-TANIS-ZIMMERMAN 9E . EXAM-STANDARD Drill the two exams, A4 in hand 手握A4,操练两场考试 Fresh problems in the MAST90105 style - probability half & inference half MAST90105风格的新题 -- probability上半与inference下半 TL;DR. Two written exams decide the unit: the mid-sem tests the probability half (transformations, MGFs, joint laws, the Poisson process) and the final tests the inference half (MoM/MLE, Fisher information & the CRLB, Bayes, pivotal CIs and hypothesis tests) with a probability tail. Each problem below names its recipe, shows the worked skeleton, and flags the trap markers punish. Drill them with your A4 sheet open - that is exactly the exam condition. TL;DR. 两场笔试决定这个单元:期中考probability上半(transformation、MGF、joint law、Poisson process),期末考 inference下半(MoM/MLE、Fisher information与CRLB、Bayes、pivotal CI与hypothesis test)外加一个概率尾部。下面每道题都点出它的recipe、展示解题骨架,并标出阅卷者会扣分的陷阱。摊开你的A4来操练它们 -- 那正是考试条件。 ★ What the exam asks here 这里考试问什么 Both written exams are A4-bring-in: ONE double-sided handwritten or printed sheet plus a non-programmable Casio FX-82, and a distribution table is provided. So your sheet should carry derivation recipes and decision logic - the MLE 3-step, the CRLB formula, the Beta-geometric update, the pivot-inversion drill, the "which statistic?" map - not a copy of the supplied table. Memorise the moves; look up the quantiles. Handy constants: z0. 975 = 1. 96, e-1 ~ 0. 368, X0. 95(11) = 19. 68, to. 95,11 = 1. 796. 两场笔试都可带 A4: 一张双面手写或打印的纸,外加一台非编程 Casio FX-82,且会提供 distribution table。所以你的纸应承载推导 recipe 与决策逻辑 -- MLE 三步法、CRLB 公式、Beta-geometric 更新、pivot 反演演练、“用哪个统计量?”的地图 -- 而不是抄一份所提供的表。把套路背下来;分位数现查。顺手的常数:,,,。 Q1 Transformation - CDF method, then NAME it via the MGF Q1变换 -- CDF方法,再用MGF给它NAME(命名) Q1 TRANSFORMATION 8 marks . W7 / final Q1 Let X have density fx(x) = 2x on (0,1). Let Y =- 2ln X. (a) Use the CDF method to find fy (y) and its support. (b) Compute My(t) and name the distribution of Y. AskSia Library · MAST90105 · 双语 Bilingual 01 Worked - CDF method first, MGF to name 1 Method: CDF method (one monotone branch), then MGF uniqueness to name the law. Y = - 2 ln X is decreasing in X, and X € (0,1) => Y € (0, 00). 方法:CDF 方法(单一单调分支),然后用 MGF uniqueness 来命名分布规律。关于是递减的,且。 2 (a) X = e-y/2, so for y > 0:[5]Source: asksia-bible-mast90105-bilingual.pdf"Estimate the parameter (" Method of Moments, then MLE (1) MOM: set E(X) =- solve ÔMM; (2) MLE: l'(0) = 0 - ÔML ; (3) check !" < 0 (or read a boundary / order statistic). "Is this the best / most efficient estimator? Find its variance" Bias, variance & the Cramér-Rao bound (1) bias: E() - 0; (2) I(0) = - E[0} In f], CRLB = 1/(nl); (3) hits the bound & unbiased => efficient (check regularity!). "Construct a confidence interval for 0" Pivotal quantity - invert (1) find a pivot V (data, 0) with a 0-free law; (2) Pr(a ≤ V ≤ b) = 1 - a; (3) invert algebraically to bracket 0. "Test Ho VS Ha / find the most powerful test" Neyman-Pearson / LRT, or a standard t / x2 / z (1) simple-vs-simple: reject when L(00)/L(01) < k; (2) set c from a (or a = B); (3) else pick the standard statistic (see D. 2) & get the p-value. "A prior distribution is given" Bayesian posterior & priorxlikelihood (1) TT(0|x)&T(0)L(0)- spot conjugacy; (2) Bayes estimate = posterior mean (squared loss) or median (absolute loss); (3) credible interval from posterior quantiles. "Do these categorical counts fit / are the factors independent?" Chi-square goodness- of-fit / contingency (1) expected E; (GOF) or Eij = RICi; (2) x2 = _ (0 - E)2/ E; (3) df = k -1 -mor (r-1)(c-1). ! Three cues that look alike but split the method 三个看着相像、却分流方法的cue (1) "Estimate" vs "best estimator" - the first wants MoM/MLE, the second wants bias / variance / CRLB. (2) "Confidence" vs "credible" interval - pivot-invert is frequentist; a prior in the stem switches you to the Bayesian posterior quantiles. (3) Test cue - "most powerful" / "simple vs simple" = Neyman-Pearson; an unspecified mean/variance/proportion = the standard t / x2 / z in D. 2. (1) “Estimate(估计)”对“best estimator(最佳估计量)”––前者要 MoM/MLE,后者要 bias / variance / CRLB。(2) "Confidence(置信)”对“credible(可信)”区间 -- pivot 反演是频率派的;题干里出现 prior (先验) 就把你切换到贝叶斯后验分位数。(3)检验线索 -- “most powerful(最强势)”/ “simple vs simple"⇒ Neyman- Pearson;一个未指明的均值/方差/比例⇒ D. 2 里的标准 t/ x2 / z。 AskSia Library · MAST90105 · 双语 Bilingual EXAM MORNING . THE DECODER EXAM MORNING . THE DECODER WHAT TO WRITE . WHAT IS ALREADY PROVIDED The bring-in A4 plan & the recipes to recall 带入的A4方案与要回想的recipe TL;DR. The exam prints the distribution table (every PMF/PDF, MGF, mean and variance for the standard families), so your one A4 double-sided sheet must carry the three things it does not. the derivation recipes, the decision logic, and the handful of off-table formulas (the Z-t-x2-F links, CRLB, the standard test statistics, the posterior updates). Below is the layout, the mid-sem/final split, and the family map - the last picture to glance at before you walk in.[11]Source: asksia-bible-mast90105-bilingual.pdfTL;DR. 考试印有distribution table(标准分布族的每个PMF/PDF、MGF、mean与variance),所以你那张A4双面纸必须装上它不给的三样东西:推导recipe、决策逻辑,以及那少数几个表外公式(Z→t→×2→F关联、CRLB、标准检验统计量、posterior更新)。下面是版面布局、期中/期末划分,以及分布族地图 -- 进考场前最后该瞥一眼的那张图。 D. 2 The standard-test selector (write these on the A4) D. 2标准检验选择器(把这些写到A4上) These statistics are not on the provided table - they are the inference engine. Pick by "what is being tested" and "is o known / is n large?" 这些统计量不在所提供的表格上 -- 它们是inference引擎。按“被检验的是什么”与“o是否已知/n是否大?”来挑。 Testing . . . Statistic Reference law Mean, o known (or large n, CLT) Z = 2 - No N(0,1) Mean, o unknown, normal data I = SVn tn-1 Variance, normal data (n-1)s2 X7-1 (one-sided, asymmetric) Proportion Z = 2- po N(0,1) VPogo/n Two variances D. 3 What goes on the A4 (and what does not) D. 3A4上写什么(以及不写什么) AskSia Library · MAST90105 · 双语 Bilingual Fm1-1,n2-1 ˉ − ✓ WRITE this (table does not give it)[13]Source: asksia-bible-mast90105-bilingual.pdfMoM & MLE; bias/variance; Fisher info & CRLB (+regularity); Bayesian posterior + credible interval; pivot Cls; Neyman-Pearson; the t / x2 / z standard tests; GOF / contingency. D. 5 The last glance: the sampling-distribution family D. 5最后一瞥:sampling-distribution分布族 AskSia Library · MAST90105 · 双语 Bilingual FIG D. 1 X2k chi-square, k df sum of k Z2 X k = 2 21 tk Student t, k df tk = Z / V(X2k / k) Z N(0, 1) F = (X2m/m) + (xªn/n) Fmin ratio, (m, n) df as df - c : tk - Z and X2 k/k + 1 (F + const) table gives each density The one map both papers lean on: a standard normal Z at the centre; square and sum k of them > X2k; Z/Vx}/k- tk; a ratio of two scaled X2 > Fm,n. The provided table gives each density; these relationships are the off- table facts that drive every CI and test. 两份试卷都依赖的那张图:中心是 standard normal Z;把它们平方求和→×2k;→ tk;两个缩放x2 之比→ Fm,no 所提供的表格给出各density;这些关系才是表上没有、却驱动每个 CI 与检验的事实。 AskSia Library · MAST90105 · 双语 Bilingual ★ Recipes to recall - the whole-course checklist 要回想的recipe -- 全课程清单 · Bayes: priorxlikelihood, renormalise by total probability. Bayes: priorxlikelihood,按 total probability 重新归一化。 · MLE: {' = 0 then !" < 0; watch for a boundary / order-statistic MLE. MOM: E(X) = x. MLE:先后;当心边界 / order-statistic MLE。MOM: 。 · CRLB: Var(0) ≥ 1/(nI(0)); regularity fails for Unif(0,0). CRLB :; 对均匀分布regularity 失效。 · Bayesian: conjugate posterior - mean (squared loss) / median (absolute loss) + credible interval. Bayesian: 共轭 posterior → mean (平方损失)/ median(绝对损失)+ credible interval。 · CI: find a pivot, Pr(a ≤ V ≤ b) = 1 - a, invert. CI:找一个 pivot,,反解。 · Tests: N-P likelihood ratio (set c from a or a = B); standard t / x2 / z (D. 2); p-value vs a. Tests: N-P likelihood ratio (由 α或设c);标准 t/ x2 / z (D. 2); p-value 对照 α。

1）Neyman–Pearson / Most powerful test

当题目是 simple vs simple，或者明确说 most powerful：
- 用 Neyman–Pearson lemma
- 拒绝域基于 likelihood ratio[5]Source: asksia-bible-mast90105-bilingual.pdf"Estimate the parameter (" Method of Moments, then MLE (1) MOM: set E(X) =- solve ÔMM; (2) MLE: l'(0) = 0 - ÔML ; (3) check !" < 0 (or read a boundary / order statistic). "Is this the best / most efficient estimator? Find its variance" Bias, variance & the Cramér-Rao bound (1) bias: E() - 0; (2) I(0) = - E[0} In f], CRLB = 1/(nl); (3) hits the bound & unbiased => efficient (check regularity!). "Construct a confidence interval for 0" Pivotal quantity - invert (1) find a pivot V (data, 0) with a 0-free law; (2) Pr(a ≤ V ≤ b) = 1 - a; (3) invert algebraically to bracket 0. "Test Ho VS Ha / find the most powerful test" Neyman-Pearson / LRT, or a standard t / x2 / z (1) simple-vs-simple: reject when L(00)/L(01) < k; (2) set c from a (or a = B); (3) else pick the standard statistic (see D. 2) & get the p-value. "A prior distribution is given" Bayesian posterior & priorxlikelihood (1) TT(0|x)&T(0)L(0)- spot conjugacy; (2) Bayes estimate = posterior mean (squared loss) or median (absolute loss); (3) credible interval from posterior quantiles. "Do these categorical counts fit / are the factors independent?" Chi-square goodness- of-fit / contingency (1) expected E; (GOF) or Eij = RICi; (2) x2 = _ (0 - E)2/ E; (3) df = k -1 -mor (r-1)(c-1). ! Three cues that look alike but split the method 三个看着相像、却分流方法的cue (1) "Estimate" vs "best estimator" - the first wants MoM/MLE, the second wants bias / variance / CRLB. (2) "Confidence" vs "credible" interval - pivot-invert is frequentist; a prior in the stem switches you to the Bayesian posterior quantiles. (3) Test cue - "most powerful" / "simple vs simple" = Neyman-Pearson; an unspecified mean/variance/proportion = the standard t / x2 / z in D. 2. (1) “Estimate(估计)”对“best estimator(最佳估计量)”––前者要 MoM/MLE,后者要 bias / variance / CRLB。(2) "Confidence(置信)”对“credible(可信)”区间 -- pivot 反演是频率派的;题干里出现 prior (先验) 就把你切换到贝叶斯后验分位数。(3)检验线索 -- “most powerful(最强势)”/ “simple vs simple"⇒ Neyman- Pearson;一个未指明的均值/方差/比例⇒ D. 2 里的标准 t/ x2 / z。 AskSia Library · MAST90105 · 双语 Bilingual EXAM MORNING . THE DECODER EXAM MORNING . THE DECODER WHAT TO WRITE . WHAT IS ALREADY PROVIDED The bring-in A4 plan & the recipes to recall 带入的A4方案与要回想的recipe TL;DR. The exam prints the distribution table (every PMF/PDF, MGF, mean and variance for the standard families), so your one A4 double-sided sheet must carry the three things it does not. the derivation recipes, the decision logic, and the handful of off-table formulas (the Z-t-x2-F links, CRLB, the standard test statistics, the posterior updates). Below is the layout, the mid-sem/final split, and the family map - the last picture to glance at before you walk in.[13]Source: asksia-bible-mast90105-bilingual.pdfMoM & MLE; bias/variance; Fisher info & CRLB (+regularity); Bayesian posterior + credible interval; pivot Cls; Neyman-Pearson; the t / x2 / z standard tests; GOF / contingency. D. 5 The last glance: the sampling-distribution family D. 5最后一瞥:sampling-distribution分布族 AskSia Library · MAST90105 · 双语 Bilingual FIG D. 1 X2k chi-square, k df sum of k Z2 X k = 2 21 tk Student t, k df tk = Z / V(X2k / k) Z N(0, 1) F = (X2m/m) + (xªn/n) Fmin ratio, (m, n) df as df - c : tk - Z and X2 k/k + 1 (F + const) table gives each density The one map both papers lean on: a standard normal Z at the centre; square and sum k of them > X2k; Z/Vx}/k- tk; a ratio of two scaled X2 > Fm,n. The provided table gives each density; these relationships are the off- table facts that drive every CI and test. 两份试卷都依赖的那张图:中心是 standard normal Z;把它们平方求和→×2k;→ tk;两个缩放x2 之比→ Fm,no 所提供的表格给出各density;这些关系才是表上没有、却驱动每个 CI 与检验的事实。 AskSia Library · MAST90105 · 双语 Bilingual ★ Recipes to recall - the whole-course checklist 要回想的recipe -- 全课程清单 · Bayes: priorxlikelihood, renormalise by total probability. Bayes: priorxlikelihood,按 total probability 重新归一化。 · MLE: {' = 0 then !" < 0; watch for a boundary / order-statistic MLE. MOM: E(X) = x. MLE:先后;当心边界 / order-statistic MLE。MOM: 。 · CRLB: Var(0) ≥ 1/(nI(0)); regularity fails for Unif(0,0). CRLB :; 对均匀分布regularity 失效。 · Bayesian: conjugate posterior - mean (squared loss) / median (absolute loss) + credible interval. Bayesian: 共轭 posterior → mean (平方损失)/ median(绝对损失)+ credible interval。 · CI: find a pivot, Pr(a ≤ V ≤ b) = 1 - a, invert. CI:找一个 pivot,,反解。 · Tests: N-P likelihood ratio (set c from a or a = B); standard t / x2 / z (D. 2); p-value vs a. Tests: N-P likelihood ratio (由 α或设c);标准 t/ x2 / z (D. 2); p-value 对照 α。

2）标准检验统计量选择器

E. Bayesian estimation

F. 分类型数据 / 非参数

1）Chi-square goodness-of-fit / contingency table

2）Distribution-free / nonparametric

1）Transformation of random variables

2）MGF

3）Bivariate / covariance / correlation

4）Poisson process

5）Sampling-distribution family map

九、如果你时间不够，最后 48 小时优先级这样排

第一优先级：final 核心五大题型

第二优先级：容易混淆的概念

第三优先级：概率尾巴

如果你愿意，我下一条可以继续直接帮你做这三件事中的任意一个：
- A：把这门课整理成“最终 A4 速记版”
- B：按章节给你列“公式 + 定义 + 高频题型”
- C：直接给你做一份“MAST90105 final 冲刺计划（3天/5天版）”
你回复我一个字母就行，我马上接着做。
（如果你愿意，我之后也可以帮你记住这门课最容易混的点，方便你下次直接快速复习。）

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