Introduction to Probability Theory

1. Introduction To Probability Theory

2. CHAPTER OneSet Theory Review Ordered Pairs and Product Sets Relations Venn diagrams Tree diagram Application of Set Theory Problems

3. CHAPTER TwoCounting Techniques • Arrangements • 2.2 Permutations • 2.3 The Multiplication Principle • 2.4 Permutation of Objects That Are Not All Different • 2.5 The Addition Principle • 2.6 Combination • 2.7 Problems

4. CHAPTER ThreeProbability • 3.1 Random Experiment & Sample Space Events • 3.2 Types of Events • 3.3 Axioms of Probability • 3.4 Problems

5. CHAPTER FourConditional Probability and Independency • 4.1 Definition Independent Events • 4.2 Conditional Probability • 4.3 Three Conditional Events or More • 4.4 Bayes Law • 4.5 Problems

6. CHAPTER FiveFrequency Distribution & Its Characteristics 5.1 Frequency distribution 5.2 Types of Class –Intervals 5.3 Exclusive and Inclusive Class-Intervals 5.4 Cumulative Relative Frequency and Cumulative istribution Function 5.5 Probability Mass Function (p.m.f.) 5.6 Probability Density Function (p.d.f.) 5.7 Cumulative Distribution Function (C.D.F.) 5.8 Problems

7. References • Appendix Tables

8. Probability Mass Function (p.m.f.) • If X is a discrete random variable (d.r.v.) which takes values x1, x2, x3, …, xn. • corresponding probabilities p1, p2, p3, …, pn. • If p(x)=x v xεR • then P(x) takes values: p1, p2, p3, …, pn respectively. P(x) will be p.m.f. of X, if:

9. ExampleFor random variable X according to the following (pmf), and (0 < p < 1), 1. Show that P(x) is p.m.f. 2. Find Pr(0≤x<1).

10. Solution Then, P(x) is a p.m.f.

11. ExampleShow that P(x) is a p.m.f. Solution • Each value of P(x) ≥ 0 Then, P(x) is a p.m.f.