Modern mathematical statistics splits into two major philosophies based on how probability is interpreted. Frequentist Statistics
(Posterior): Updated belief about the parameter after observing data.
Frequentist statistics treats parameters as fixed, unknown constants. Bayesian statistics treats parameters as random variables with their own probability distributions. Bayes' Theorem for Inferences
An unbiased estimator that achieves this lower bound is called . Methods of Finding Estimators mathematical statistics lecture
The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true. If the p-value is less than or equal to a predetermined significance level ( , usually 0.05), the null hypothesis is rejected. 6. Advanced Statistical Frameworks
P(θ|data)=P(data|θ)P(θ)P(data)cap P open paren theta vertical line data close paren equals the fraction with numerator cap P open paren data vertical line theta close paren cap P open paren theta close paren and denominator cap P open paren data close paren end-fraction
Take on uncountable values within an interval (e.g., human height). Modeled using a Probability Density Function (PDF). Expectation, Variance, and Moments Expected Value ( ): The long-run average or mean value of a random variable. Variance ( If the p-value is less than or equal
To review a mathematical statistics lecture effectively, you should focus on the that connects probability to data analysis . Unlike introductory statistics, mathematical statistics is primarily proof-based and focuses on developing statistical rules rather than just applying them. Core Lecture Components
limn→∞P(|θ̂−θ|<ϵ)=1for any ϵ>0limit over n right arrow infinity of cap P open paren the absolute value of theta hat minus theta end-absolute-value is less than epsilon close paren equals 1 space for any epsilon is greater than 0 Common Methods for Finding Estimators Method of Moments (MoM)
To create rigorous mathematical frameworks to quantify the uncertainty of these inferences. 5. Hypothesis Testing Instead
Uses Student's t-distribution, which accounts for the extra uncertainty of estimating the variance from a small sample. 5. Hypothesis Testing
Instead, the 95% confidence level refers to the . If we repeat the experiment infinitely many times and calculate a confidence interval each time, 95% of those calculated intervals will contain the true population parameter. Deriving a CI for the Population Mean ( ) with Known Variance ( σ2sigma squared By the Central Limit Theorem, the sample mean X̄cap X bar follows a normal distribution: