1.2 Random variables

Random variables are the building blocks of probability theory and statistical analysis in Bayesian statistics. They represent numerical outcomes of random processes, allowing us to quantify uncertainty and model real-world phenomena.

Understanding random variables is crucial for making probabilistic inferences. We'll explore their types, properties, and common distributions, as well as how they're used in Bayesian analysis for parameter estimation, hypothesis testing, and prediction.

Definition of random variables

• Random variables form the foundation of probability theory and statistical analysis in Bayesian statistics
• These variables represent numerical outcomes of random processes or experiments, allowing for quantitative analysis of uncertainty
• Understanding random variables is crucial for modeling real-world phenomena and making probabilistic inferences

Discrete vs continuous variables

• Discrete random variables take on countable, distinct values (number of customers in a store)
• Continuous random variables can assume any value within a given range (temperature, height)
• Discrete variables use probability mass functions while continuous variables use probability density functions
• Mixed random variables combine both discrete and continuous components (insurance claim amounts)

Probability mass functions

• Describe the probability distribution for discrete random variables
• Assign probabilities to each possible outcome of the discrete random variable
• Must satisfy two key properties: non-negative probabilities and sum to 1
• Represented mathematically as $P(X = x) = f_X(x)$ for a discrete random variable X

Probability density functions

• Characterize the probability distribution for continuous random variables
• Represent the relative likelihood of a continuous random variable taking on a specific value
• Area under the curve between two points gives the probability of the variable falling within that range
• Defined mathematically as $f_X(x) = \frac{d}{dx}F_X(x)$ where $F_X(x)$ is the cumulative distribution function

Properties of random variables

• Properties of random variables provide essential information about their behavior and characteristics
• These properties help in summarizing and comparing different random variables
• Understanding these properties is crucial for making inferences and predictions in Bayesian statistics

Expected value

• Represents the average or mean value of a random variable over many repetitions
• Calculated as the sum of each possible value multiplied by its probability for discrete variables
• For continuous variables, computed as the integral of the product of the variable and its probability density function
• Denoted as $E[X] = \mu$ and serves as a measure of central tendency

Variance and standard deviation

• Variance measures the spread or dispersion of a random variable around its expected value
• Calculated as the expected value of the squared deviation from the mean: $Var(X) = E[(X - \mu)^2]$
• Standard deviation is the square root of variance, providing a measure of spread in the same units as the original variable
• Both variance and standard deviation are crucial for assessing the uncertainty and variability in random variables

Moments and moment-generating functions

• Moments provide a way to characterize the shape and properties of probability distributions
• First moment corresponds to the expected value, second central moment to the variance
• Higher-order moments describe skewness (3rd) and kurtosis (4th) of the distribution
• Moment-generating functions uniquely determine the probability distribution of a random variable
• Used to derive moments and other properties of random variables efficiently

Common probability distributions

• Probability distributions describe the likelihood of different outcomes for random variables
• These distributions play a crucial role in modeling various phenomena in Bayesian statistics
• Understanding common distributions helps in selecting appropriate models for different scenarios

Discrete distributions

• Bernoulli distribution models binary outcomes (success/failure) with probability p
• Binomial distribution represents the number of successes in n independent Bernoulli trials
• Poisson distribution models the number of events occurring in a fixed interval (time or space)
• Geometric distribution describes the number of trials until the first success in repeated Bernoulli trials

Continuous distributions

• Normal (Gaussian) distribution characterized by its bell-shaped curve and symmetric properties
• Uniform distribution assigns equal probability to all values within a specified range
• Exponential distribution models the time between events in a Poisson process
• Gamma distribution generalizes the exponential distribution and is often used in Bayesian analysis

Multivariate distributions

• Joint normal (multivariate Gaussian) distribution extends the normal distribution to multiple variables
• Dirichlet distribution serves as a multivariate generalization of the beta distribution
• Multinomial distribution generalizes the binomial distribution to multiple categories
• Wishart distribution models covariance matrices in multivariate Bayesian analysis

Transformations of random variables

• Transformations allow for manipulation and analysis of random variables in different forms
• These techniques are essential for deriving new distributions and solving complex probabilistic problems
• Understanding transformations helps in adapting existing models to specific research questions

Linear transformations

• Involve scaling and shifting random variables: $Y = aX + b$
• Preserve many properties of the original distribution, including normality
• Affect the mean and variance of the random variable predictably
• Commonly used in standardization and normalization of data

Non-linear transformations

• Include operations like squaring, taking logarithms, or applying trigonometric functions
• Can significantly alter the shape and properties of the original distribution
• Often used to model complex relationships or to satisfy assumptions in statistical analyses
• Require careful consideration of how the transformation affects the probability distribution

Jacobian method

• Technique for finding the probability density function of a transformed random variable
• Involves calculating the determinant of the Jacobian matrix of partial derivatives
• Essential for deriving distributions of functions of random variables
• Applies to both univariate and multivariate transformations

Joint and conditional distributions

• Joint and conditional distributions describe relationships between multiple random variables
• These concepts are fundamental to understanding dependencies and making inferences in Bayesian statistics
• Crucial for modeling complex systems with interrelated variables

Marginal distributions

• Obtained by summing or integrating out other variables from a joint distribution
• Provide information about individual variables without considering others
• Calculated using the law of total probability for discrete variables
• For continuous variables, involve integrating the joint probability density function

Conditional probability

• Describes the probability of an event given that another event has occurred
• Calculated using the formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Forms the basis for Bayesian inference and updating beliefs based on new information
• Allows for incorporating prior knowledge and updating probabilities with observed data

Independence of random variables

• Two random variables are independent if knowledge of one does not affect the probability of the other
• For independent variables, $P(A \cap B) = P(A)P(B)$ and $f_{X,Y}(x,y) = f_X(x)f_Y(y)$
• Independence simplifies many calculations and is often assumed in statistical models
• Testing for independence is crucial in many applications of Bayesian statistics

Functions of random variables

• Functions of random variables allow for modeling complex relationships and deriving new distributions
• These concepts are essential for many statistical techniques and probabilistic modeling
• Understanding functions of random variables is crucial for advanced applications in Bayesian statistics

Sum of random variables

• Involves adding two or more random variables to create a new random variable
• For independent variables, the mean of the sum equals the sum of the means
• Variance of the sum of independent variables is the sum of their variances
• Convolution is used to find the distribution of the sum of continuous random variables

Product of random variables

• Results from multiplying two or more random variables
• Often encountered in modeling ratios, areas, or volumes
• For independent variables, $E[XY] = E[X]E[Y]$ but this doesn't hold for dependent variables
• Distribution of the product can be complex, often requiring special techniques to derive

Ratio of random variables

• Involves dividing one random variable by another
• Commonly used in modeling rates, proportions, or relative measures
• Can lead to challenging distributions, especially if the denominator can be close to zero
• Cauchy distribution arises as the ratio of two independent standard normal random variables

Bayesian perspective on random variables

• Bayesian statistics treats parameters as random variables with probability distributions
• This approach allows for incorporating prior knowledge and updating beliefs based on data
• Understanding the Bayesian perspective is crucial for applying Bayesian methods in statistical analysis

Prior distributions

• Represent initial beliefs or knowledge about parameters before observing data
• Can be informative (based on previous studies) or non-informative (minimal assumptions)
• Common priors include conjugate priors which simplify posterior calculations
• Selection of priors is a crucial step in Bayesian analysis and can influence results

Likelihood functions

• Describe the probability of observing the data given specific parameter values
• Treated as a function of the parameters with fixed observed data
• Play a central role in both frequentist and Bayesian statistics
• In Bayesian analysis, combined with the prior to form the posterior distribution

Posterior distributions

• Represent updated beliefs about parameters after observing data
• Calculated using Bayes' theorem: $P(\theta|D) \propto P(D|\theta)P(\theta)$
• Combine information from the prior distribution and the likelihood function
• Serve as the basis for Bayesian inference, parameter estimation, and prediction

Sampling from random variables

• Sampling techniques are essential for generating random numbers from specific distributions
• These methods are crucial for Monte Carlo simulations and Bayesian computation
• Understanding sampling techniques is important for implementing Bayesian algorithms

Inverse transform sampling

• Generates samples from any probability distribution given its cumulative distribution function
• Involves applying the inverse of the CDF to uniform random variables
• Works well for distributions with closed-form inverse CDFs (exponential, uniform)
• Can be computationally expensive for distributions without closed-form inverse CDFs

Rejection sampling

• Generates samples from a target distribution using a proposal distribution
• Accepts or rejects samples based on a comparison with the target distribution
• Useful for sampling from complex or multimodal distributions
• Efficiency depends on how closely the proposal distribution matches the target

Importance sampling

• Estimates properties of a target distribution using samples from a different distribution
• Assigns weights to samples to correct for the difference between the sampling and target distributions
• Particularly useful in Bayesian inference for approximating posterior expectations
• Can be more efficient than rejection sampling for certain types of problems

Applications in Bayesian inference

• Bayesian inference applies probability theory to statistical problems
• This approach allows for updating beliefs based on new evidence and quantifying uncertainty
• Understanding these applications is crucial for implementing Bayesian methods in practice

Parameter estimation

• Involves estimating unknown parameters of a statistical model using observed data
• Bayesian estimation provides a full posterior distribution rather than point estimates
• Allows for incorporating prior knowledge and quantifying uncertainty in estimates
• Common estimators include posterior mean, median, and mode (MAP estimate)

Hypothesis testing

• Bayesian hypothesis testing compares the relative evidence for different hypotheses
• Uses Bayes factors to quantify the strength of evidence in favor of one hypothesis over another
• Allows for comparing non-nested models and incorporating prior probabilities of hypotheses
• Provides a more nuanced approach to hypothesis testing than traditional p-values

Prediction and forecasting

• Bayesian prediction involves making probabilistic statements about future observations
• Utilizes the posterior predictive distribution to account for parameter uncertainty
• Allows for incorporating multiple sources of uncertainty in forecasts
• Particularly useful in fields like finance, weather forecasting, and epidemiology