Probability & Random Variables Statistics

Probability & Random Variables

1. Sri Lanka Institute of Advanced Technological Education (SLIATE) Advanced Technological Institute - Kurunegala Higher National Diploma in Information Technology Batch - 2 3 2 4 (FT) Probability & Binomial Distributions Group C

2. Introduction to Probability & Random Variable Understanding Uncertainty Probability is the mathematical framework for analyzing chance events and quantifying uncertainty. A Random Variable is a quantity whose value is determined by the outcome of a probability experiment. These concepts are fundamental tools across diverse fields: Finance & Risk Assessment Scientific Research Engineering & Quality Control Data Science & Machine Learning ‹#›

3. Types of Random Variables Classification of Random Variables Random variables are classified based on the nature of their possible values: Discrete Random Variables Continuous Random Variables Take on countable number of distinct values Take on any value within a given range Values can be counted Values are measured Examples: Number of heads in coin flips, count of defective items Examples: Height, weight, temperature, time Represented by Probability Mass Function (PMF) Represented by Probability Density Function (PDF) ‹#›

4. Probability Distribution Table Understanding Probability Distributions A probability distribution is a mathematical function that describes the probability of different possible values of a random variable. For a discrete random variable , we can represent its distribution using a Probability Mass Function (PMF) : x (outcome) P(X = x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Example: Fair six-sided die roll (sum of probabilities = 1) ‹#›

5. Expected Value – Concept & Importance What is Expected Value? The Expected Value (E[X] or μ) of a random variable is the long-run average outcome if the random experiment were repeated many times. It represents the center of mass of a probability distribution – a weighted average where each possible value is weighted by its probability. Importance of Expected Value: Provides a single representative value for a distribution Essential for decision-making under uncertainty Fundamental in risk assessment and financial analysis Basis for comparing different random variables ‹#›

6. Expected Value – Formula & Example Formula for Discrete Random Variables For a discrete random variable X with possible values x₁, x₂, ..., xₙ and corresponding probabilities P(X = x₁), P(X = x₂), ..., P(X = xₙ): E[X] = Σ [x · P(X = x)] Step-by-step calculation: Identify all possible values of the random variable Determine the probability of each value Multiply each value by its probability Sum all products to get the expected value The expected value represents the long-run average outcome if the random experiment were repeated many times. ‹#›

7. Variance & Standard Deviation Measuring Spread in Random Variables Variance (Var(X)) measures the spread or dispersion of a random variable's distribution around its mean. Var(X) = E[(X - μ X ) 2 ] Standard Deviation (SD(X)) is the square root of variance, providing a measure of spread in the original units of the random variable. SD(X) = σ X = √Var(X) A high variance indicates widely dispersed values, while a low variance suggests values clustered closely around the mean. These measures are critical for understanding risk and variability in random processes. ‹#›

8. Introduction to Binomial Distribution What is a Binomial Distribution? The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials. Each trial is called a Bernoulli trial , which has only two possible outcomes: Success (with probability p) Failure (with probability 1-p) Common examples of binomial scenarios: Number of heads in multiple coin flips Number of defective items in a sample Number of successful medical treatments Number of correct answers on a multiple-choice test ‹#›

9. Assumptions / Conditions of Binomial Distribution Four Key Requirements For a random variable to follow a binomial distribution, these four conditions must be met: 1. Fixed Number of Trials (n ) 2. Two Possible Outcomes 3. Independent Trials 4. Constant Probability of Success (p) If any of these conditions are not met, the binomial distribution may not be an appropriate model for the random variable. ‹#›

10. Binomial Formula & Example The Binomial Probability Formula For a binomial random variable X with parameters n (number of trials) and p (probability of success): P(x) = n C x · p x (1 − p) n−x P(X = k): Probability of exactly k successes C(n,k): Number of ways to get k successes in n trials p k : Probability of k successes (1-p) n-k : Probability of (n-k) failures ‹#›

11. Mean, Variance & Shape of Binomial Distribution Properties of the Binomial Distribution For a binomial random variable X with parameters n (number of trials) and p (probability of success): Mean (Expected Value): E[X] = np Variance: Var(X) = np(1-p) Standard Deviation: SD(X) = √[np(1-p)] Shape of the distribution: Symmetric when p = 0.5 Right-skewed when p < 0.5 Left-skewed when p > 0.5 Approaches normal distribution as n increases ‹#›

12. Applications & Conclusion Key Takeaways Summary of Key Concepts: Probability quantifies uncertainty in random events Random variables map outcomes to numerical values Probability distributions describe the likelihood of all possible values Expected value represents the long-run average outcome Variance and standard deviation measure the spread of distributions Binomial distribution models the number of successes in fixed trials These concepts provide a powerful framework for analyzing uncertainty and making informed decisions across numerous fields including finance, medicine, engineering, and data science. ‹#›

13. References García-García, J. I., Fernández-Lebrón, M. M., Fernández-Feria, M. I., & Berenguer, M. I. (2022). The Binomial Distribution: Historical Origin and Evolution of Its Problem Situations. Mathematics , 10(15), 2680. https://doi.org/10.3390/math10152680 González, J., Tuerlinckx, F., & De Boeck, P. (2016). A Note on the Poisson's Binomial Distribution in Item Response Theory. Applied Psychological Measurement , 40(4), 302–310. https://doi.org/10.1177/0146621616629380 Kozak, K. (n.d.). Chapter 5: Discrete Probability Distributions - Section 5.1. Coconino Community College . Retrieved October 8, 2025, from https://www.coconino.edu/resources/files/pdfs/academics/sabbatical-reports/kate-kozak/chapter_5.pdf Pishro-Nik, H. (2014). Introduction to probability, statistics, and random processes . Kappa Research LLC. Available at: https://www.probabilitycourse.com ScienceDirect. (n.d.). Binomial Distribution - an overview . Retrieved October 8, 2025, from https://www.sciencedirect.com/topics/engineering/binomial-distribution ScienceDirect. (n.d.). Random Variable - an overview . Retrieved October 8, 2025, from https://www.sciencedirect.com/topics/mathematics/random-variable Shafer, D. S., & Zhang, Z. (2023). The Binomial Distribution . In Introductory Statistics. LibreTexts. Retrieved October 8, 2025, from https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_(Shafer_and_Zhang)/04%3A_Discrete_Random_Variables/4.03%3A_The_Binomial_Distribution Skellam, J. G. (1948). A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials. Journal of the Royal Statistical Society: Series B (Methodological) , 10(2), 257-261. https://doi.org/10.1111/j.2517-6161.1948.tb00013.x Taufiq, I., Miatun, A., & Anggoro, B. S. (2020). Binomial distribution at high school: An analysis based on the mathematical connection. Journal of Physics: Conference Series , 1521(3), 032087. https://doi.org/10.1088/1742-6596/1521/3/032087 Vega, E. (n.d.). 3.3 The Binomial Formula . Portland Community College. Retrieved October 8, 2025, from https://spot.pcc.edu/~evega/section-14.html Viti, A., Terzi, A., & Bertolaccini, L. (2015). A practical overview on probability distributions. Journal of Thoracic Disease , 7(3), E7-E10. https://doi.org/10.3978/j.issn.2072-1439.2015.01.3 7 ‹#›