Mathematical Statistics Lecture File

Course Overview Mathematical Statistics is the bridge between probability theory and real-world data. While probability asks, "Given a process, what is the likelihood of an outcome?" statistics asks the inverse: "Given the outcome, what can we infer about the process?"

The relationship between Type I and Type II errors is a trade-off. Decreasing ( \alpha ) (making the test stricter) increases ( \beta ) (missing a real effect). 6. Practical Worked Example (Normal Data) Problem: A machine fills cereal boxes with mean weight 500g. A sample of 10 boxes yields: ( \barx = 492g ), sample standard deviation ( s = 9g ). Is the machine underfilling? (Assume ( \alpha = 0.05 )). mathematical statistics lecture

| Property | Definition | Mathematical Condition | | :--- | :--- | :--- | | | On average, you hit the target. | ( \mathbbE[\hat\theta] = \theta ) | | Consistency | As sample size ( n \to \infty ), ( \hat\theta \to \theta ). | ( \lim_n\to\infty P(|\hat\theta - \theta| > \epsilon) = 0 ) | | Efficiency | Minimal variance among unbiased estimators. | ( \textVar(\hat\theta) \leq \textVar(\tilde\theta) ) for any other unbiased ( \tilde\theta ) | The Cramér-Rao Lower Bound (CRLB): There is a physical limit to how small the variance can be. [ \textVar(\hat\theta) \geq \frac1n \mathbbE\left[\left(\frac\partial \log f(x;\theta)\partial \theta\right)^2\right] ] If an estimator achieves the CRLB, it is called efficient . 4. Interval Estimation: Quantifying Uncertainty A point estimate ( \hat\theta = 3.2 ) is useless without error bounds. A Confidence Interval (CI) gives a range that covers ( \theta ) with a prescribed probability ( 1-\alpha ). Constructing a CI for the Mean (( \sigma ) known) Assume ( X_i \sim \mathcalN(\mu, \sigma^2) ). We know that: [ Z = \frac\barX - \mu\sigma / \sqrtn \sim \mathcalN(0, 1) ] Is the machine underfilling