Probability and Combinatorics

Contents

1. Permutations and Combinations

1.1 Introduction     

1.2 Principles of counting

1.3 Permutations     

1.4 Combinations

2. Integer Solutions of Linear Equations

2.1 Introduction     

2.2 Integer solutions of linear equations    

2.3 Binomial identities    

3. Principle of Inclusion and Exclusion

3.1 Introduction     

3.2 Principle of inclusion and exclusion     

3.3 Derangements     

3.4 Generating functions

4. Recurrence Relations    

4.1 Introduction     

4.2 The process of obtaining a recurrence relation 

4.3 The homogeneous equation     

4.4 The non-homogeneous equation 

4.5 Pigeonhole principle   

5. Probability and Probability Distributions

5.1 Introduction     

5.2 Deterministic and non deterministic models

5.3 Sample space     

5.4 Events     

5.5 Probability

5.6 Random variables 

5.7 Continuous random variables

6. Special Probability Distributions

6.1 Introduction     

6.2 Discrete distributions

6.3 Continuous distributions     

7. Expectation

7.1 Introduction     

7.2 Expectation of a discrete random variable

7.3 Expectation of continuous random variable

8. Moment and Cumulant Generating Functions

8.1 Moment Generating Functions (M.G.F.)

8.2 Cumulant Generating Function (C.G.F.)          

8.3 Characteristic functions     

8.4 M(T) and K(T) for some common distributions

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