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5.17. Probability Calculation (Mandatory)

5.17.1. Justification ↑ Back to top

Probability and statistics are fundamental in computer science for algorithm analysis, system modeling, decision-making under uncertainty, and data analysis. This course integrates probability theory with statistical methods, focusing on computational applications such as algorithm analysis, machine learning, system modeling, and data science.

5.17.2. Generales Goals ↑ Back to top

1. Understand the fundamentals of probability and statistical inference.
2. Apply probability distributions and statistical methods to solve computing problems.
3. Develop skills for modeling stochastic systems and performing data inference.

5.17.3. Contribution to Outcomes ↑ Back to top

AG-C07) Computing Knowledge: Applies knowledge of mathematics, science, and computing. (Familiarity)

5.17.4. Content ↑ Back to top

5.17.4.1. Probability Fundamentals (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Probability spaces and axioms
2. Event operations: union, intersection, complement
3. Counting techniques: permutations and combinations
4. Conditional probability and independence
5. Bayes' theorem and applications

Learning Outcomes

1. Define sample spaces and apply probability axioms. [Familiarity]
2. Use counting techniques to calculate probabilities. [Usage]
3. Apply Bayes' theorem in classification problems. [Assessment]

5.17.4.2. Discrete Random Variables (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Discrete random variables
2. Probability mass function (PMF) and cumulative distribution function (CDF)
3. Expectation, variance, and moments
4. Distributions: Bernoulli, Binomial, Geometric, Poisson
5. Applications in computational system modeling

Learning Outcomes

1. Define and characterize discrete random variables. [Familiarity]
2. Calculate expectation and variance for different distributions. [Usage]
3. Apply discrete distributions in network and system modeling. [Assessment]

5.17.4.3. Continuous Random Variables (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Continuous random variables
2. Probability density function (PDF) and CDF
3. Transformations of random variables
4. Distributions: Uniform, Exponential, Normal
5. Applications in system simulation and queuing theory

Learning Outcomes

1. Differentiate between discrete and continuous variables. [Familiarity]
2. Calculate probabilities using density functions. [Usage]
3. Model service times and arrival times using continuous distributions. [Assessment]

5.17.4.4. Multivariate Distributions and Dependence (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Joint and marginal distributions
2. Conditional distributions
3. Covariance and correlation
4. Independence of random variables
5. Applications in multidimensional data analysis

Learning Outcomes

1. Calculate joint and marginal distributions. [Familiarity]
2. Measure dependence using covariance and correlation. [Usage]
3. Analyze relationships between variables in datasets. [Assessment]

5.17.4.5. Limit Theorems and Approximations (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Law of Large Numbers
2. Central Limit Theorem
3. Moments and generating functions
4. Probabilistic inequalities (Chernoff, Markov, Chebyshev)
5. Applications in big data and algorithm analysis

Learning Outcomes

1. State and interpret limit theorems. [Familiarity]
2. Apply CLT in distribution approximations. [Usage]
3. Use inequalities in algorithm bound analysis. [Assessment]

5.17.4.6. Bayesian Inference (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Ross, 2014; Devore, 2016)

Topics

1. Bayesian updating with discrete/continuous priors
2. Maximum A Posteriori (MAP) estimation
3. Bayesian credible intervals
4. Conjugate priors
5. Applications in Naive Bayes classifiers and ML

Learning Outcomes

1. Differentiate between frequentist and Bayesian approaches. [Familiarity]
2. Perform Bayesian belief updating. [Usage]
3. Implement MAP estimation in machine learning problems. [Assessment]

5.17.4.7. Statistical Inference (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Devore, 2016)

Topics

1. Parameter estimation: MLE (Maximum Likelihood Estimation)
2. Hypothesis testing: z-test, t-test, chi-square test
3. Confidence intervals
4. Bootstrapping and resampling methods
5. Applications in model validation and data science

Learning Outcomes

1. Estimate parameters using maximum likelihood methods. [Familiarity]
2. Perform hypothesis testing to validate assumptions. [Usage]
3. Construct confidence intervals and apply bootstrapping. [Assessment]

5.17.4.8. Regression and Advanced Applications (6 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Devore, 2016)

Topics

1. Simple linear regression
2. Least squares fitting
3. Poisson processes
4. Monte Carlo simulation
5. Introduction to Markov chains
6. Applications in prediction and system modeling

Learning Outcomes

1. Implement linear regression models. [Familiarity]
2. Use Monte Carlo simulation to solve complex problems. [Usage]
3. Apply stochastic processes in real system modeling. [Assessment]

5.17.5. Bibliography ↑ Back to top

Ross, S. M. (2014). A First Course in Probability. Pearson.

Devore, J. L. (2016). Probability and Statistics for Engineering and the Sciences. Cengage Learning.