Mathematical Statistics — Lecture Better

The final pillar of our lecture is hypothesis testing. This is the formal procedure for deciding between two competing claims: the null hypothesis and the alternative hypothesis. We use a test statistic to determine if the observed data is sufficiently extreme to warrant rejecting the null hypothesis. This process involves a delicate balance between Type I errors (false positives) and Type II errors (false negatives). The p-value, perhaps the most famous metric in statistics, tells us the probability of obtaining results at least as extreme as the ones observed, assuming the null hypothesis is true.

(Prior): Initial belief about the parameter before seeing data. (Likelihood): The data-generating mechanism. mathematical statistics lecture

𝜕𝜕θlnL(θ)=0the fraction with numerator partial and denominator partial theta end-fraction l n cap L open paren theta close paren equals 0 5. Interval Estimation (Confidence Intervals) The final pillar of our lecture is hypothesis testing

Today’s lecture is about , and the professor—a wiry woman with a taste for dramatic pauses—poses a question that sounds like a Zen koan: “Given that you have seen the data, what is the most plausible story the universe could be telling you?” This process involves a delicate balance between Type

Equate the population moments to the sample moments and solve for the parameters.

This is where the notation becomes dense. You will see:

p(θ|x)=f(x|θ)π(θ)∫f(x|θ)π(θ)dθp open paren theta vertical line x close paren equals the fraction with numerator f of open paren x vertical line theta close paren pi open paren theta close paren and denominator integral of f of open paren x vertical line theta close paren pi open paren theta close paren d theta end-fraction