MATH 350: Probability
Final Exam Review Topics
December 13, 2024
1. Pseudo-random numbers; U(0, 1) and related distributions
2. Approximating integrals (areas) and constants with the use of random numbers (Monte-Carlo method)
3. Sample spaces; events; discrete and continuous random variables; finite and countably infinite probability distributions of discrete random variables
4. Basic theorems about probability
5. Tree probability diagrams
6. Continuous random variables; density functions; cumulative distribution functions
7. Uniform and exponential continuous random variables
8. Obtaining one- and multi-dimensional probability distributions and density functions of random variables based on the U(0, 1) distribution with the use of geometry
9. The Birthday Problem
10. Permutations and k-permutations
11. Multiplication principle of combinatorics
12. Binomial coefficients; binomial theorem
13. Combinations; applications in card games
14. Bernoulli process and binomial distribution; practical applications; "at least" and "at most" cases
15. Geometric distribution
16. Discrete conditional probabilities
17.
18. Bayes' probabilities and Bayes' formula; applications
19. Joint discrete probability distributions
20.
21. Memory-less property of the exponential distribution; the "Bus Paradox"
22. Continuous joint distributions; joint continuous densities
23. Marginal densities; independence of joint continuous random variables
24. Poisson distribution and its derivation
25. Important continuous densities: uniform, exponential, normal (standard and non-standard)
26. Functions of random variables - derivation of density using the CDF method
27. Expected value of discrete and continuous random variables and functions of random variables
28. Expected value of important discrete and continuous random variables
29. Expected value of sums and products of random variables
30. Variance and standard deviation of discrete and continuous random variables; properties of variance; variance of sum of random variables; variance of average
31. The method of convolution and its use in finding distributions and densities of sums of discrete and continuous random variables
32. Chebyshev Inequality; derivation; applications
33. Estimating
accuracy of a
34. (Weak) Law of Large Numbers for discrete and continuous random variables; derivation of the law
35. Standardizing random variables
36. Standardized sums and several versions of Central Limit Theorem (Theorems 9.2, 9.4, 9.5, and 9.6)
37. The meaning of Central Limit Theorem
38. Using Z-table; "continuity correction"
39. Moments and moment generating functions for most important discrete and continuous distributions; derivations
40. The moment series
41. Theorems 10.2 and 10.5 about the moment generating function
42. Proof of the Central Limit Theorem (Theorem 9.6)
43. Covariance and correlation coefficient for discrete and continuous random variables
44. Introduction to random walks