1. Poisson Distribution and Hypothesis Testing: Modeling shoe sales with Poisson distribution, computing exact and compound probabilities, and testing for sales increase using significance levels.

2. Discrete Probability and Sampling Distribution: Constructing and analyzing the sampling distribution of mean and mode from combinations of 20p and 50p coins, determining underlying probabilities.

3. Binomial Probability and Significance Testing: Using binomial distribution to assess bounce test failures, hypothesis testing for process improvement, and calculating minimum sample size for significance.

4. Continuous Distribution and Expectation Transformation: Working with a defined PDF to compute probabilities, mean, variance, and transforming expectations in geometrical contexts (area of circles).

5. Poisson Process and Normal Approximation: Calculating defect probabilities in steel cable, applying binomial distributions across multiple samples, and using normal approximation to estimate production length.

6. Cumulative Distribution Analysis: Solving for constants in a piecewise-defined cumulative distribution function using given expectation and variance values, applying algebraic manipulation and integration.