Daily Derivation Exercises

EE 364: Spring 2026

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

Derive all the BEG-CUP pdf moments: binomial, geometric, hypergeometric, negative binomial, Poisson, gamma, exponential, chi-square, beta, uniform, Gaussian.

Week 9

Derive the pdf for \(Y = g(X)\) if \(g(x)\) is 1-to-1.

Week 10

Prove the population Uncertainty Principle (Cauchy-Schwarz): \(\sigma_{XY}^2 \le \sigma_X^2 \sigma_Y^2\).

Week 11

State the formal definition for all UC-MOPED convergences. Prove the Weak Law of Large Numbers: \(\overline{X}_n \xrightarrow{p} \mu_X\).

Week 12

Show that the sample variance is unbiased and consistent: \(E[S_X^2(n)] = \sigma_X^2\) for all \(n\) and \(S_X^2(n) \xrightarrow{p} \sigma_X^2\).

Week 14

Prove the Central Limit Theorem: \(Z_n = \text{STD}(\overline{X}_n) \xrightarrow{d} Z \sim N(0,1)\).

Week 15

Derive all MGFs on the BEG-CUP pdf sheet.