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Prompt

Act as an expert LaTeX coder and a university-level statistics professor. Generate the complete, compilable LaTeX source code for an ultra‑compact, A4 landscape, 4‑column micro‑cheat‑sheet. **SYLLABUS TO COVER (Strictly adhere to these units):** - UNIT I: Probability – Random experiments, sample space, events, algebra of events. Classical, statistical, and axiomatic definitions. Laws of addition and multiplication. Independence and conditional probability. - UNIT II: Random Variables – Discrete and continuous. Probability mass function (p.m.f), probability density function (p.d.f), cumulative distribution function (c.d.f) and its properties. Univariate transformations with illustrations. - UNIT III: Expectation – Properties and moments. Moment generating function (m.g.f) and characteristic function (c.f). - UNIT IV: Standard Distributions – Binomial, Poisson, Geometric, Uniform, Normal, Exponential, and Gamma. Their properties (mean, variance, m.g.f, additivity, memorylessness, approximations). **QUESTION PATTERN TO FOLLOW:** - **Group B (Short Answers):** 20 questions, each carrying **5 marks**. Answers must be concise but mathematically complete. - **Group C (Long Answers):** 10 questions, each carrying **10 marks**. Answers must include full derivations, proofs, or numerical illustrations. **AVOID DUPLICATION (DO NOT use these previously asked topics):** Do NOT generate questions on: definitions of equally likely/independent/mutually exclusive events, basic definition of probability/p.m.f, additive theorem, Bayes' theorem, basic Binomial mean/variance, basic marginal/conditional probability, basic Random variable/expectation, general Discrete vs Continuous, basic Poisson mean/variance, general Normal distribution importance, or the B⊂A subset proof. Instead, generate questions covering the **remaining/new topics** listed below. **SPECIFIC NEW TOPICS FOR 5-MARK QUESTIONS (20 total):** 1. Law of total probability – statement and a numerical example. 2. Pairwise independence vs. mutual independence – define and explain why pairwise does not imply mutual (give a counterexample concept). 3. Markov's inequality – state and prove it. 4. Chebyshev's inequality – state and prove it. 5. Covariance and correlation – define and list at least 3 key properties. 6. Hazard function (failure rate) – define mathematically and give the Exponential special case. 7. Memorylessness of Geometric distribution – prove it step-by-step. 8. Standard normal distribution and standardization – define Z and show how to standardize any X~N(μ,σ²). 9. Sum of independent normal variables – state the property (mean and variance of the sum). 10. Gamma distribution – define its p.d.f, state its mean and variance. 11. Additivity of Gamma distributions – state the property and prove it using m.g.f.s. 12. Derive the m.g.f. of Exponential(λ) – show the integral and final result. 13. Derive the characteristic function of Exponential(λ) – show the integral. 14. Prove that Binomial(n,p) is symmetric when p=0.5. 15. Poisson approximation to Binomial – state the approximation, conditions (n large, p small, np=λ), and a rule of thumb for when to use it. 16. Expectation of a function (LOTUS) – define for both discrete and continuous cases. 17. Variance properties for dependent vs. independent variables – state Var(aX+bY) general formula and simplify for independence. 18. Inclusion–exclusion principle for three events – state the full formula. 19. Relationship between covariance and independence – explain why zero covariance does NOT imply independence, give a brief counterexample (e.g., X~N(0,1), Y=X²). 20. Univariate transformation of a continuous random variable – state the CDF method and the Jacobian (change-of-variable) formula. **SPECIFIC NEW TOPICS FOR 10-MARK LONG QUESTIONS (10 total):** 1. State and prove the law of total probability with a detailed numerical example (e.g., using a partition of 3 events). 2. Derive the mean and variance of the Exponential distribution from its p.d.f. using integration by parts. 3. Derive the mean and variance of the Gamma(α, λ) distribution using the Gamma function properties. 4. State and prove Chebyshev's inequality. Discuss its significance and limitations compared to the empirical rule. 5. Prove the memorylessness property for both the Exponential and Geometric distributions. 6. State the properties of the moment generating function (M_X(0)=1, uniqueness, derivatives for moments, and sum of independent variables). Prove at least two of these properties. 7. Discuss univariate transformations: explain the CDF technique and the change-of-variable technique for continuous variables with an illustration (e.g., X~Exp(1), find distribution of Y=√X). 8. Derive the Poisson approximation to the Binomial distribution rigorously (showing the limit of the Binomial p.m.f as n→∞, p→0, np→λ). 9. Define the cumulative distribution function (c.d.f). State its four fundamental properties (monotonicity, limits, right-continuity, probability calculation) and prove that for a continuous variable, P(a<X≤b)=F(b)-F(a). 10. Define covariance and correlation. Derive the formula for Var(X+Y) in terms of variances and covariance, and extend it to Var(aX+bY). Show how independence simplifies the result. **FORMATTING REQUIREMENTS (Strictly ultra-compact, landscape, 4-column):** - Document class: `\documentclass[8pt,a4paper,landscape]{extarticle}` - Margins: `left=0.35cm, right=0.35cm, top=0.35cm, bottom=0.35cm` - Use `multicols*{4}` for columns. - Use `colorbox` for section headings (gray) and question backgrounds (lighter gray). - Define a `\SQ` command for short questions (increments counter, prints [5] marks, answer in blue). - Define an `\LQ` command for long questions (increments counter, prints [10] marks, answer in blue). - Set `\fontsize{5.15pt}{5.55pt}\selectfont` for body text. - `\parskip`, `\linespread`, and `\setlist` must be ultra-dense (micro cheat sheet style). - Include all necessary packages (amsmath, amssymb, enumitem, xcolor, titlesec, microtype, etc.). **OUTPUT INSTRUCTION:** Output ONLY the complete LaTeX code inside a single code block. Do not include any explanations before or after the code. The code must compile without errors on Overleaf or any TeX Live distribution. The answers must be mathematically correct, self-contained, and formatted perfectly within the tight space constraints.