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:
Your current (e.g., undergraduate student, graduate researcher, working data scientist) The specific textbook or curriculum you are following
, emphasize that the course is proof-heavy and may not use real data at all. The "Best" Estimator:
Before analyzing data, we must define the mathematical "ground rules." Statistics relies on Measure Theory mathematical statistics lecture
These lecture notes are a living document. For deeper understanding, work through derivations and solve problems—statistics is learned by doing .
An estimator is consistent if it converges in probability to the true parameter value as the sample size increases. Interval Estimation (Confidence Intervals)
Among unbiased estimators, the one with the smallest variance is the most efficient. The Cramer-Rao Lower Bound provides the theoretical minimum variance for any unbiased estimator. Instead, the 95% confidence level refers to the
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The MLE is the parameter value that maximizes the likelihood function, meaning it makes the observed data most probable.Given a joint probability density function , the likelihood function is:
An unbiased estimator that achieves this lower bound is called . Methods of Finding Estimators The "Best" Estimator: Before analyzing data, we must
: Measures the average squared difference between the estimator and the parameter. It decomposes cleanly into variance and bias:
To understand the value of the lecture, you must first distinguish Mathematical Statistics from its cousins.
In a standard applied stats class, the professor gives you a formula (e.g., ( \barx = \frac\sum x_in )) and tells you when to use it. In a , the professor asks: Why is ( \barx ) the best guess for the population mean? What does 'best' even mean mathematically?
In engineering, medicine, and data science, we rarely have access to an entire population. Instead, we work with a subset. Mathematical statistics bridges the gap between this observed data and the unobserved truth. 2. Sample Space, Random Variables, and Population
) : The status quo, representing no effect, no difference, or no change. : The claim the researcher wants to establish. Step 2: Errors in Testing