📊Mathematical Modeling Unit 7 – Probability and Statistics

Key Concepts and Definitions

• Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1
• Random variables assign numerical values to outcomes of a random experiment
  • Discrete random variables have countable outcomes (number of heads in 10 coin flips)
  • Continuous random variables have an infinite number of possible outcomes within a range (height of students in a class)
• Probability distributions describe the likelihood of different outcomes for a random variable
• Population refers to the entire group of individuals or items being studied
• Sample is a subset of the population used to make inferences about the whole population
• Descriptive statistics summarize and describe the main features of a data set
• Inferential statistics use sample data to make predictions or draw conclusions about a population

Probability Basics

• Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes
• Mutually exclusive events cannot occur at the same time (rolling a 1 and a 6 on a die)
• Independent events do not influence each other's probability (flipping a coin and rolling a die)
• Conditional probability is the probability of an event occurring given that another event has already occurred
• The addition rule states that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• The multiplication rule for independent events states that $P(A \cap B) = P(A) \times P(B)$
• Bayes' theorem describes the probability of an event based on prior knowledge of related conditions

Types of Probability Distributions

• Binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes (pass/fail)
• Poisson distribution models the number of events occurring in a fixed interval of time or space (number of customers arriving per hour)
• Normal distribution is a continuous probability distribution that is symmetric and bell-shaped
  • Characterized by its mean $\mu$ and standard deviation $\sigma$
  • Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three
• Exponential distribution models the time between events in a Poisson process (time between customer arrivals)
• Uniform distribution has equal probability for all outcomes within a given range (selecting a random number between 1 and 10)
• Other distributions include gamma, beta, and chi-square, used in specific modeling situations

Descriptive Statistics

• Measures of central tendency describe the center of a data set
  • Mean is the average value, calculated by summing all values and dividing by the number of observations
  • Median is the middle value when data is ordered from least to greatest
  • Mode is the most frequently occurring value
• Measures of dispersion describe the spread of a data set
  • Range is the difference between the maximum and minimum values
  • Variance measures the average squared deviation from the mean, calculated as $\frac{\sum(x_i - \bar{x})^2}{n-1}$
  • Standard deviation is the square root of the variance
• Skewness measures the asymmetry of a distribution (positive skew has a longer right tail)
• Kurtosis measures the tailedness of a distribution (higher kurtosis indicates more outliers)
• Percentiles divide data into 100 equal parts (50th percentile is the median)

Inferential Statistics

• Hypothesis testing evaluates claims about a population based on sample data
  • Null hypothesis ($H_0$) assumes no significant difference or effect
  • Alternative hypothesis ($H_a$) proposes a significant difference or effect
• P-value is the probability of obtaining the observed results if the null hypothesis is true
  • A small p-value (typically < 0.05) suggests strong evidence against the null hypothesis
• Confidence intervals estimate a population parameter using sample data and a level of confidence (95% CI means 95% chance the true value falls within the interval)
• T-tests compare means between two groups or to a hypothesized value
• ANOVA (analysis of variance) compares means between three or more groups
• Regression analysis models the relationship between variables
  • Linear regression fits a straight line to predict a dependent variable from one or more independent variables
  • Logistic regression predicts a binary outcome based on one or more predictors

Data Visualization Techniques

• Scatter plots display the relationship between two continuous variables
• Line graphs show trends or changes over time
• Bar charts compare categorical data using rectangular bars
• Histograms display the distribution of a continuous variable using bins
• Box plots summarize the five-number summary (minimum, Q1, median, Q3, maximum) and identify outliers
• Heatmaps use color intensity to represent values in a matrix
• Pie charts show proportions of a whole, but can be difficult to interpret accurately

Applications in Mathematical Modeling

• Queuing theory models waiting lines and service systems (bank tellers, call centers)
• Markov chains model systems that transition between states with fixed probabilities (weather patterns, animal population dynamics)
• Monte Carlo simulations use random sampling to estimate complex probabilities or optimize decisions (stock prices, project management)
• Bayesian inference updates probabilities based on new evidence (medical diagnosis, spam filters)
• Stochastic differential equations model systems with random noise (population growth, financial markets)
• Machine learning algorithms use statistical models to make predictions or decisions (image recognition, recommendation systems)

Advanced Topics and Extensions

• Multivariate analysis examines relationships between multiple variables simultaneously
  • Principal component analysis (PCA) reduces dimensionality while preserving variation
  • Cluster analysis groups similar observations based on their characteristics
• Time series analysis models data collected over regular time intervals (stock prices, weather data)
  • Autoregressive (AR) models use past values to predict future values
  • Moving average (MA) models use past forecast errors to predict future values
• Survival analysis models time-to-event data with censoring (customer churn, medical studies)
• Bayesian networks represent probabilistic relationships between variables using directed acyclic graphs
• Bootstrapping resamples data to estimate sampling distributions and confidence intervals
• Markov Chain Monte Carlo (MCMC) methods sample from complex probability distributions (parameter estimation, model selection)