Daily Derivation Exercises

Week 1
- Prove the Addition Theorem: \(P(A \cup B) + P(A \cap B) = P(A) + P(B)\).
Week 2
- Prove the Theorem of Total Probability and Bayes Theorem for evidence \(E\) and a partition of hypotheses \(H_1, \ldots, H_n\).
Week 3
- Prove the Binomial Theorem.
Week 4
- Prove the Trinomial Theorem.
- Derive the trinomial PMF \(p(x, y)\).
- Show that the marginal \(p(x)\) and conditional \(p(x \mid y)\) are both binomial.
Week 5
- Prove the Poisson Law: \(b(n, p) \rightarrow P(\lambda)\) if \(\lambda = n p\).
Week 6
- Prove \(\Gamma(\alpha + 1) = \alpha \, \Gamma(\alpha)\) for \(\alpha > 0\). Then show \(\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}\).
Week 8
Week 9
Week 10
Week 11
Week 12
Week 13
Week 14

Week 1