Expectation

• What Is the Expectation of a Function Symmetric About Zero?
  A function symmetric about zero is even, so its expectation is not generally zero; symmetry makes expectations of integrable odd functions vanish.
• What Is the Expected Maximum of Two Exponential Variables?
  For independent exponential variables with rates λ₁ and λ₂, the expected maximum is 1/λ₁ + 1/λ₂ − 1/(λ₁ + λ₂).
• How Does Expected Value Change with the Probability Measure?
  Expected value depends on the probability measure: integrate the same random variable under each measure, or use a density to convert between them.
• How to Calculate E[XY] for Dependent Random Variables
  For dependent random variables, compute E[XY] from their joint distribution or use covariance; the marginal means alone are not enough.
• How Do You Sum a Random Number of Independent Variables?
  For an independent random count and iid summands, derive the random sum’s mean and variance, see a worked example, and learn when the formulas fail.