Probability Distribution

Course Title: Probability Distribution

Course No: BDS153

Nature of the Course: Theory + Lab

Semester: 2

Full Marks: 45 + 30

Pass Marks: 18 + 12

Credit Hours: 3

Course Description

This course focuses on probability distribution related to fundamental statistical techniques that are of particular relevance to Data Science. The course covers random variables, mathematical expectation, discrete and continuous probability distributions, bivariate distributions and their applications.

Course Objectives

Describe about probability distribution along with essential characteristics and properties - Calculate and describe mathematical expectation and its properties.
Describe the characteristics features of and compute probabilities using discrete and continuous probability distributions.
Describe the bivariate distributions and calculate moments of bivariate distributions, conditional expectations, conditional variance.
Use Jacobian of transformations for bivariate distributions.
To apply these techniques in different scenario.

Course Contents

1. Random Variables

8 hrs

1.1. Random variables and their properties

1.2. types of random variables: qualitative (Categorical) and quantitative, discrete and continuous random variables

1.3. Probability distribution of a random variable

1.4. probability mass function

1.5. probability density function and its properties

1.6. functions of random variable

1.7. Transformation of random variable

1.8. Joint probability mass function

1.9. joint probability distribution function

1.10. joint probability density function

1.11. conditional probability mass function

1.12. conditional probability density function

1.13. Solving Data Science related numerical problems

2. Mathematical Expectation

8 hrs

2.1. Mathematical expectation of a random variable

2.2. properties of mathematical expectation

2.3. addition and multiplicative theorems of expectation

2.4. covariance and correlation

2.5. conditional expectation

2.6. conditional variance

2.7. variance of linear combination of random variables

2.8. Moments of random variables

2.9. raw and central moments

2.10. Generating functions: moment generating function

2.11. characteristic function

2.12. probability generating function

2.13. cumulant generating function with their properties

2.14. Solving Data Science related numerical problems

3. Discrete Probability Distribution

10 hrs

3.1. Binomial distribution

3.2. Poisson distribution

3.3. negative binomial distribution

3.4. geometric distribution

3.5. hyper-geometric distribution

3.6. negative hyper-geometric distribution: their mass functions, distribution functions, moments, moment generating functions, characteristic functions and properties, distribution fittings

3.7. Solving Data Science related numerical problems

4. Continuous Probability Distribution

12 hrs

4.1. Uniform distribution

4.2. Normal distribution

4.3. Lognormal distribution

4.4. Exponential distribution

4.5. Beta distribution

4.6. Gamma distribution

4.7. Weibull distribution, their density functions, distribution functions, moments, moment generating functions, characteristic functions, properties and uses with distribution fittings

4.8. Solving Data Science related numerical problems

5. Bivariate Distribution

10 hrs

5.1. Bivariate distribution for discrete and continuous variables: joint, marginal, and conditional distributions

5.2. independence of random variables

5.3. Transformations of random variables: Jacobian of transformations

5.4. distributions of sum, product and ratio of random variables

5.5. Solving Data Science related numerical problems

Laboratory Works

The laboratory work includes using any statistical software such as Microsoft Excel, SPSS, R etc.

1.Random number generation, expectation, variance, covariance, raw and central moments
2.Calculation of probabilities using Binomial distribution, Poisson distribution, negative binomial distribution and fitting of distributions
3.Calculation of probabilities using normal distribution, lognormal distribution, exponential distribution, and fitting of distributions
4.Calculation of descriptive statistics of probability distribution

Reference Books

1.Blitzstein Joseph K. & Hwang Jessica (2015). Introduction to Probability, CRC Press
2.Baron Michael (2014). Probability and Statistics for Computer Scientists, CRC Press
3.Gupta S. C. and Kapoor V. K. (2007). Fundamentals of Mathematical Statistics, Sultan Chand and Sons
4.Rohatgi V. K. and Ehsanes Saleh, A. K. MD (2005). An Introduction to Probability and Statistics, John Wiley & Sons
5.Bruce Peter and Bruce Andrew (2017). Practical Statistics for Data Scientists, O'Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472, U.S.A.

Notes:

This syllabus follows the official BDS curriculum of Tribhuvan University. In case of any doubt or revision, the university's published syllabus shall be considered authoritative.

Probability and Statistics