STA260 Summer 2024

About These Notes

Tutorial notes written by Anna Ly and Ting Lin for STA260 during the Summer 2024 semester. Although it's called "Probability and Statistics II", a more appropriate name would be "an introduction to mathematical statistics" based on the course description, content, and textbook used.

The instructor was Luai Al Labadi and we followed Mathematical Statistics with Applications (7th Edition) by Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer.

Some questions were picked from past tests and exams from varying instructors and are high in difficulty.

Tutorial Notes

Tutorial 1

Topics: Sample variance, sampling distributions related to the normal distribution such as the \(\chi^{2}\) distribution.

Worksheet ↓ Interesting Remark ↓ Solutions: Q1 Q2 Q3

Tutorial 2

Topics: Properties of sampling distributions related to the normal distribution, central limit theorem, normal approximation to the binomial distribution.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4

Note: There is an issue with the answer for Q3 — you are NOT supposed to claim \(P(\bar{Y} \leq y) = P(Z \leq z)\) as if they are equal. The correct approach uses the approximation via the CLT.

Tutorial 3

Topics: Biased and unbiased estimators.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4

Tutorial 4

Topics: Pivotal quantity, confidence intervals.

Worksheet ↓ Solutions: Q1 Q2 Q3

Note: The reason for picking \(a = 0.02\) is because you are focusing on the support of \(w\), which is \(0 < w < 1\). If you were to pick \(a > 1\), then \(P(W \leq a) = 1\), which does not provide what we want. Also note: when using statistical tables for the \(Z, \chi^{2}_{n}, t_{v}, F_{n_1,n_2}\) distributions, the textbook uses upper-tail values (\(\alpha/2\)), whereas the CDF convention uses lower-tail values (\(1 - \alpha/2\)). R is consistent with the CDF convention.

Tutorial 5

Topics: Confidence intervals using distributions related to the normal distribution, computing variance of sample variance.

Worksheet ↓ Solutions: Q1 Q2 Q3

Tutorial 6

Topics: Relative efficiency, consistency.

Worksheet ↓ Solutions: Q1 Q2 (Sol. 1) Q2 (Sol. 2) Q3 (Sol. 1) Q3 (Sol. 2)

Note: There is a rendering issue for the second solution of Q3 — the question is the same in the worksheet regardless.

Tutorial 7

Topics: Sufficiency, completeness, Rao-Blackwell theorem.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4

Note: There is a slight typo for Q2 — it should cite the fundamental theorem of calculus as \(= h'(x)\), not \(= h(x)\).

Tutorial 8

Topics: Cramér-Rao inequality, exponential family.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4

Note: There is a small typo in the worksheet — Fisher's Information uses the partial derivative \(\partial\), not the univariate derivative \(d\). These are not the same.

Tutorial 9

Topics: Maximum likelihood estimators, method of moments estimators.

Worksheet ↓ Solutions: Q1 Q2 Q3

Tutorial 10

Topics: Hypothesis testing using the NP-lemma and likelihood ratio tests.

Worksheet ↓ Solutions: Q1 Q2 Q3

Note: Typo in Q2 final solution — \(\sigma^{2}\) should be 9, not 3; multiply the last rejection region value by 9. Typo in Q3 — should be \(\chi^{2}_{(1)}\) instead of \(\chi^{2}_{(16)}\); the numerical value is associated with 1 degree of freedom.

Tutorial 11 & 12

Final exam review.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4 Q5

Cancelled Tutorial — Simple Linear Regression

Previous iterations of the course covered simple linear regression, but there was not enough time this semester. It is typically re-taught in STA302. Worksheets are posted here for reference.

Worksheet ↓ Solutions: Q1 Q2 Q3 Q4

Additional Guides

Formula / Review Sheet

Created for the final exam during the Winter 2022 session. A good summary of most equations and formulas you should know before term tests and exams.

Formula Sheet ↓

Textbook Solution Manual

A public solution manual for Mathematical Statistics with Applications (7th Ed.). Not official — there are plenty of typos, and solutions may not be detailed enough for full marks on assessments. Useful for getting directions when lost.

Solution Manual ↗

Guides for Using Statistical Tables

Guides for the statistical tables in Mathematical Statistics with Applications (7th Ed.).

\(Z\) Table ↓ \(t_v\) Table ↓ \(F_{n_1,n_2}\) Table ↓ \(\chi^2_n\) Table ↓

STA260 — Probability & Statistics II

About These Notes

Tutorial Notes

Additional Guides