12.2 Random Variables and Expectation

Random variables are the building blocks of probability theory, assigning numerical values to random events. They come in two flavors: discrete (countable outcomes) and continuous (any value in a range). Understanding random variables is crucial for analyzing uncertain phenomena.

Probability distributions describe the likelihood of outcomes for random variables. For discrete variables, we use probability mass functions (PMFs) to assign probabilities to each possible value. These tools help us model real-world scenarios and make predictions about uncertain events.

Random Variables

Types of Random Variables

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• Random variable assigns numerical values to outcomes of random experiments
• Discrete random variable takes on countable number of distinct values (dice rolls, number of customers)
• Continuous random variable assumes any value within a specified range (height, weight, temperature)
• Probability distribution describes likelihood of each possible outcome for a random variable

Properties and Applications

• Random variables enable mathematical analysis of uncertain events
• Used in various fields including statistics, physics, and finance
• Can be represented graphically using probability distributions
• Facilitate calculation of probabilities for complex events
• Allow for modeling of real-world phenomena with inherent randomness

Probability Distributions

Probability Mass Function

• Probability mass function (PMF) defines probability distribution for discrete random variables
• Assigns probabilities to each possible value of the discrete random variable
• Must satisfy two conditions:
  1. Probabilities are non-negative: $P(X = x) \geq 0$ for all x
  2. Sum of all probabilities equals 1: $\sum_{x} P(X = x) = 1$
• Represented graphically as a bar chart or stem plot
• Used to calculate probabilities of specific outcomes or ranges of outcomes
• Cumulative distribution function (CDF) derived from PMF by summing probabilities

Examples and Applications

• Binomial distribution models number of successes in fixed number of independent trials (coin flips)
• Poisson distribution describes number of events occurring in fixed interval (customer arrivals)
• Geometric distribution represents number of trials until first success (attempts to win a game)
• Hypergeometric distribution models sampling without replacement (drawing cards from a deck)
• PMFs help analyze discrete phenomena in various fields (quality control, reliability engineering)

Expectation and Variance

Expected Value and Its Properties

• Expected value represents average outcome of a random variable over many trials
• Calculated as sum of each possible value multiplied by its probability: $E[X] = \sum_{x} x P(X = x)$
• Provides measure of central tendency for probability distribution
• Linearity of expectation states:
  1. $E[aX + b] = aE[X] + b$ for constants a and b
  2. $E[X + Y] = E[X] + E[Y]$ for random variables X and Y
• Used in decision-making, risk assessment, and financial modeling

Measures of Variability

• Variance quantifies spread of random variable around its expected value
• Calculated as expected value of squared deviations from mean: $Var(X) = E[(X - E[X])^2]$
• Alternative formula: $Var(X) = E[X^2] - (E[X])^2$
• Standard deviation equals square root of variance: $\sigma = \sqrt{Var(X)}$
• Provides measure of dispersion in same units as random variable
• Chebyshev's inequality relates standard deviation to probability of deviations from mean
• Variance and standard deviation used in statistical inference, risk management, and quality control