🔀Stochastic Processes Unit 2 – Random Variables and Distributions

Key Concepts

• Random variables map outcomes of random experiments to numerical values
• Probability distributions describe the likelihood of different values occurring for a random variable
• Cumulative distribution functions (CDFs) give the probability that a random variable takes a value less than or equal to a given value
• Probability mass functions (PMFs) and probability density functions (PDFs) characterize the probability distribution for discrete and continuous random variables, respectively
• Expected value represents the average value of a random variable over many trials
• Variance and standard deviation measure the spread or dispersion of a random variable's values around its expected value
• Independence and conditional probability play crucial roles in analyzing multiple random variables

Types of Random Variables

• Discrete random variables take on a countable set of distinct values (integers, finite sets)
• Continuous random variables can take on any value within a specified range or interval
• Mixed random variables have both discrete and continuous components in their probability distribution
• Bernoulli random variables have only two possible outcomes, typically labeled as success (1) and failure (0)
• Binomial random variables count the number of successes in a fixed number of independent Bernoulli trials
• Poisson random variables model the number of events occurring in a fixed interval of time or space, given a constant average rate

Probability Distributions

• Probability distributions assign probabilities to the possible values of a random variable
• Discrete probability distributions are characterized by probability mass functions (PMFs)
  • PMFs give the probability of a random variable taking on each possible value
  • The sum of all probabilities in a PMF must equal 1
• Continuous probability distributions are characterized by probability density functions (PDFs)
  • PDFs describe the relative likelihood of a random variable taking on different values
  • The area under the PDF curve between two values represents the probability of the random variable falling within that range
• Joint probability distributions describe the probabilities of multiple random variables occurring together
• Marginal probability distributions are obtained by summing or integrating joint distributions over the values of other variables

Important Properties

• Expected value (mean) is the weighted average of a random variable's possible values, weighted by their probabilities
  • For discrete random variables, $E[X] = \sum_{x} x \cdot P(X=x)$
  • For continuous random variables, $E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx$
• Variance measures the average squared deviation of a random variable from its expected value
  • $Var(X) = E[(X - E[X])^2]$
  • Variance can also be calculated as $Var(X) = E[X^2] - (E[X])^2$
• Standard deviation is the square root of the variance and has the same units as the random variable
• Covariance measures the linear relationship between two random variables
  • Positive covariance indicates variables tend to increase or decrease together
  • Negative covariance indicates variables tend to move in opposite directions
• Correlation coefficient is a standardized measure of the linear relationship between two random variables, ranging from -1 to 1

Common Distributions

• Normal (Gaussian) distribution is characterized by its bell-shaped curve and is determined by its mean and standard deviation
• Uniform distribution assigns equal probability to all values within a specified range
• Exponential distribution models the time between events in a Poisson process or the lifetime of an object with a constant failure rate
• Gamma distribution is a generalization of the exponential distribution and models waiting times until a specified number of events occur
• Beta distribution is defined on the interval [0, 1] and is often used to model probabilities or proportions
• Chi-square, t, and F distributions are used in statistical inference and hypothesis testing

Transformations and Operations

• Linear transformations of a random variable $Y = aX + b$ result in changes to the expected value and variance
  • $E[Y] = aE[X] + b$
  • $Var(Y) = a^2Var(X)$
• Functions of random variables create new random variables with their own probability distributions
  • The distribution of the function can be derived using the change of variables technique or the cumulative distribution function method
• Convolution is used to find the distribution of the sum of two independent random variables
  • For discrete random variables, the PMF of the sum is the convolution of the individual PMFs
  • For continuous random variables, the PDF of the sum is the convolution of the individual PDFs
• Moment-generating functions (MGFs) and characteristic functions uniquely characterize probability distributions and simplify calculations involving sums of independent random variables

Applications in Stochastic Processes

• Random variables are fundamental building blocks in stochastic processes, which model systems that evolve randomly over time
• Markov chains use discrete random variables to represent the states of a system and transition probabilities between states
• Poisson processes model the occurrence of events over time using Poisson random variables for the number of events in a given interval
• Brownian motion is a continuous-time stochastic process that models random movements and is characterized by normally distributed increments
• Queueing theory relies on random variables to analyze waiting times, service times, and the number of customers in a queueing system
• Stochastic differential equations incorporate random variables to model the unpredictable fluctuations in dynamic systems

Problem-Solving Techniques

• Identify the type of random variable (discrete, continuous, or mixed) and its probability distribution
• Use the cumulative distribution function (CDF) to calculate probabilities and quantiles
• Apply the probability mass function (PMF) or probability density function (PDF) to find the likelihood of specific values or ranges
• Utilize expected value, variance, and other properties to characterize and compare random variables
• Recognize common probability distributions and their key features to simplify problem-solving
• Break down complex problems into simpler components, such as independent random variables or conditional probabilities
• Apply transformations and operations, such as linear transformations or convolutions, to derive the distributions of new random variables
• Use moment-generating functions (MGFs) or characteristic functions to simplify calculations and determine the distribution of sums of independent random variables