Mathematical Statistics Lecture

The air in the lecture hall was thick with the scent of old chalk and the quiet desperation of eighty undergraduates. At the front, Professor Aris stood before a blackboard already half-covered in the cryptic runes of mathematical statistics.

• Pros: Simple, intuitive, often consistent.
• Cons: Not always the most efficient; may produce "impossible" estimates (e.g., estimating a probability $>1$).

• Discrete RV: Takes countable values (e.g., number of heads, Poisson count).
• Continuous RV: Takes values in an interval (e.g., height, time, temperature).

• Transformations of Random Variables: Using the CDF method or the Jacobian method for bivariate transformations.
• Moment Generating Functions (MGFs): Uniqueness theorem and finding distributions of sums.
• Multivariate Distributions: Marginal, conditional, and copulas.

• Invariance Property: If $\hat\theta$ is the MLE of $\theta$, then $g(\hat\theta)$ is the MLE of $g(\theta)$.
• Asymptotic Normality: For large $n$, the MLE is approximately normally distributed around the true parameter.

3. Point Estimation

The core problem: We want to find a "good" statistic to estimate $\theta$. We call this statistic an Estimator, denoted $\hat\theta$. mathematical statistics lecture