7.1 Properties of expectation

Expectation of random variables is a key concept in probability theory. It measures the average or central value of a probability distribution, helping us understand the long-term behavior of random processes and make informed decisions under uncertainty.

Calculating expected values differs for discrete and continuous variables. For discrete variables, we sum the products of each outcome and its probability. For continuous variables, we integrate the product of each value and its probability density function.

Expectation of Random Variables

Definition and Basic Properties

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• Expectation of a random variable (E[X]) measures average or central value of probability distribution
• Calculated differently for discrete and continuous random variables
• Linear operator with property E[aX + b] = aE[X] + b, where a and b are constants
• Expected value of a constant equals the constant itself (E[c] = c)
• Expectation of sum of random variables equals sum of individual expectations (E[X + Y] = E[X] + E[Y])
• May not exist for all random variables (heavy-tailed distributions)

Applications and Considerations

• Represents center of mass for probability distribution
• Expected value may not be a possible outcome of random variable
• Used in various fields (finance, statistics, physics)
• Helps analyze long-term behavior of random processes
• Crucial in decision-making under uncertainty
• Forms basis for more advanced statistical concepts (variance, covariance)

Expected Value for Discrete Variables

Calculation Methods

• Probability Mass Function (PMF) assigns probabilities to each possible value
• Formula for discrete random variable X: $E[X] = \sum(x * P(X = x))$
• Summation taken over all possible values for finite discrete variables
• Extends to infinity for infinite discrete variables (Poisson distribution)
• Multiply each outcome by its probability and sum products
• Simplified formulas exist for well-known distributions (Binomial, Poisson)

Examples and Applications

• Coin toss (Heads = 1, Tails = 0): E[X] = 1 * 0.5 + 0 * 0.5 = 0.5
• Dice roll: E[X] = 1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6) = 3.5
• Binomial distribution (n trials, probability p): E[X] = np
• Poisson distribution (rate λ): E[X] = λ
• Used in gambling to calculate expected winnings
• Applied in insurance to determine fair premiums

Expected Value for Continuous Variables

Calculation Methods

• Probability Density Function (PDF) describes relative likelihood of values
• Formula for continuous random variable X: $E[X] = \int(x * f(x) dx)$
• Integration performed over entire support of random variable
• Simplified formulas for well-known distributions (Normal, Exponential)
• Numerical integration techniques used for complex PDFs
• Represents center of mass of probability distribution

Examples and Applications

• Uniform distribution on [a, b]: E[X] = (a + b) / 2
• Exponential distribution (rate λ): E[X] = 1 / λ
• Normal distribution (mean μ, standard deviation σ): E[X] = μ
• Used in physics to calculate expected position of particles
• Applied in finance to estimate future stock prices
• Utilized in engineering to predict system performance

Linearity of Expectation

Fundamental Principles

• States E[aX + bY] = aE[X] + bE[Y], where X and Y are random variables, a and b are constants
• Holds true for both independent and dependent random variables
• Extends to finite number of variables: E[X1 + X2 + ... + Xn] = E[X1] + E[X2] + ... + E[Xn]
• Simplifies calculations for expected value of sums or linear combinations
• Applies to both discrete and continuous random variables
• Often easier than directly calculating distribution of sum

Applications and Problem-Solving

• Analyzing algorithms (average-case time complexity)
• Gambling problems (expected total winnings from multiple games)
• Financial modeling (portfolio expected returns)
• Network analysis (expected total traffic in a system)
• Queueing theory (expected waiting times in service systems)
• Combinatorial problems (expected number of successes in multiple trials)