Probability theory and its applications

Major Topics:

1. Introduction                                                                                                                             

          1-1- The concept of probability

           1-2- A review on theory of sets

           1-3- Statistical experiment

           1-4- Sample space and its several types

           1-5- Probability definition and its axioms

           1-6- Probability definition and its axioms

           1-7- Members enumeration of sample space

           1-8- Conditional probability and independent events

           1-9- Partition, the law of total probability and Bayes theorem

2. Random Variables and probability distributions

           2-1- Definition of random variables

           2-2- Several types of random variables

           2-3- Probability distribution and its types

           2-4- Multidimensional random variables
           2-5- Multivariate probability distributions

3. Mathematical expectation

           3-1- Definition of mathematical expectation

           3-2- Expected value of several types of random variables

           3-3- Expected value of functions of random variables

           3-4- Variance, Covariance and Coefficient of corolation

           3-5- Moments of random variable

           3-6- Moment generating function

           3-7- Properties of mathematical expectation

           3-8- Conditional expectation

4. Discrete probability distributions

           4-1- discrete uniform distribution

           4-2- Bernoulli and binomial distributions

           4-3- Multinomial distribution

           4-4- Hypergeometric distribution

           4-5- Geometric distribution

           4-6- Negative binomial distribution

           4-7- Poisson distribution

5. Continuous probability distribution

          5-1- Continuous uniform distribution

          5-2- Gamma distribution

          5-3- Exponential distribution

          5-4- Chi-Squared distribution

          5-5- Beta distribution

          5-6- Normal distribution

6. Functions of random variables  

          6-1- Finding random variables distributions

          6-2- Method of cumulative distribution function

          6-3- Method of transformation

7. Limiting distributions and theorems

          7-1- Chebyshev inequality

          7-2- Central limit theorem

Prescribed Text:

      1- Akhavan Niaki  S. Taghi,  "Probability theorem and its application", (In Persian)

      2- Ross Sheldon "A first course in probability"