Probability & Statistics

This course closes the sequence of mathematical courses. In this course we emphasize and illustrate the use of probabilistic models and statistical methodology that is employed in countless applications in all areas of science and engineering.

Prerequisite(s): It is necessary to be familiar with the concepts of Calculus to learn elementary probability theory and properties of distribution of random variables. In additional, a modest amount of a matrix algebra is used to support the linear regression models, Markov chains and Stochastic simulations.

Prerequisite courses: Calculus 1 (Mandatory), Linear Algebra (Mandatory)

• Fundamental counting rule

• Permutations P(n,k)

• Combinations C(n,k)

• Theoretical probability

• Additional & Multilication Rules

• Conditional probability

• Law of total probabilities

• Bayes' theorem

• Random Variables

• Probability Mass function

• Cumulative Distribution function

• Expected Value

• Variance

• Bernoulli trial

• Binomial distribution

• Poisson distribution

• Poisson vs Exponential Distribution

• Normal Distribution

• Intro to Exponential distribution

• Exponential vs Poisson

• Cumulative distribution, Expected value & variance

• Memory-less distributions

• Density function of a new variable defined as a function of another variable

• Joint distributions.

• Marginal distributions.

• Independent variables.

• Covariance.

• Convolution of distributions

• Generating random values

• the Law of Large Numbers

• Central Limit Theorem

• Density function of sum of two variables

• Expected values of sum of two variables

• Intro to Statistics;

• Characteristics of data;

• Common sampling methods;

• Statistical graphs;

• measures of center: mean, median and mode

• measures of variation: variance, standard deviation, range

• Inferences about population parameter

• Likelihood function, log-likelihood function

• Maximum likelihood estimates

• Likelihood functions in continuous case

• level of confidence, critical values

• error margin

• confidence intervals for mean (big-samples)

• t-distribution

• confidence intervals for mean (small-samples)

• confidence intervals for variance

• chi-squared distribution

• null hypothesis & alternative hypothesis

• P-value & regection regions

• Test statistics for population mean, proportion and variance