Probability

模块目标

Compute with discrete and continuous models while keeping the underlying space, σ-algebra language at syllabus grain, and independence hypotheses explicit.

课程大纲

1. 01 Probability spaces 20h

   学习结果 Check Kolmogorov axioms on a finite space and compute a derived event.

   形成性检查 四选一检查

2. 02 Conditional probability and independence 20h

   学习结果 Apply Bayes or the law of total probability; give an example that is uncorrelated but not independent.

   形成性检查 反例

3. 03 Discrete random variables 20h

   学习结果 Derive a pmf, expectation, and variance for a named discrete family.

   形成性检查 书面推导

4. 04 Continuous random variables 20h

   学习结果 Use a density and a cdf, and say when a density is not a probability.

   形成性检查 四选一检查

5. 05 Expectation and variance 20h

   学习结果 Prove linearity of expectation without independence and derive the covariance identity for Var(X+Y); independence is sufficient, not necessary, for variance additivity.

   形成性检查 书面推导

6. 06 Joint distributions 15h

   学习结果 Factor a joint pmf/pdf under independence and compute a marginal.

   形成性检查 书面推导

7. 07 Limit theorems 20h

   学习结果 State LLN and CLT with hypotheses; refuse a finite-n certainty claim.

   形成性检查 四选一检查

8. 08 Synthesis 15h

   学习结果 Model a short experiment end to end: space, variable, expectation, approximation caveat.

   形成性检查 书面推导

学习结果

Specify a probability space for a named experiment, compute a distribution or expectation, and apply a limit theorem with its hypotheses.

学科技能

Axioms, conditional probability, discrete and continuous laws, expectation, joint laws, LLN/CLT at undergraduate grain.

可迁移技能

Name the model before quoting a probability; distinguish independence from pairwise uncorrelatedness.