Expected Value

Definition

The Expected Value $E[X]$ is the long-term average value of repetitions of the same experiment it represents.

• How to read: “The expected value of X.”
• Meaning: The probability-weighted center of a random variable—long-run mean over repeated trials.

Why It Matters

The expected value is the “center of mass” for a probability distribution, providing the fundamental link between the rules of a game and its long-run average outcome. It is the essential “pivot point” for decision theory and finance, allowing us to value risk and predict the results of repeated experiments with mathematical precision.

Core Concepts

• Discrete Case: For a random variable $X$ with outcomes $x_i$ and probabilities $p_i$: $E[X] = \sum_{i=1}^n x_i p_i$

  • How to read: “The expected value of X equals the sum from i equals one to n of x i times p i.”
  • Meaning: Probability-weighted average outcome—long-run mean if you repeat the experiment many times.

• Continuous Case: For a probability density function $p(x)$: $E[X] = \int_{-\infty}^\infty x p(x) dx$

  • How to read: “The expected value of X equals the integral from negative infinity to infinity of x times p of x dx.”
  • Meaning: Same weighted average, but outcomes form a continuum—integrate x against the density.

• Linearity of Expectation: $E[aX + bY] = aE[X] + bE[Y]$, regardless of whether $X$ and $Y$ are independent.

  • How to read: “The expected value of the quantity a X plus b Y equals a times the expected value of X plus b times the expected value of Y.”
  • Meaning / when to use: Expectation distributes over linear combinations—powerful for sums of many random variables even when correlated.

• Vector Expected Value: For a random vector $X$, $E[X]$ is the vector of the expected values of its components.

  • How to read: “The expected value of bold X equals the vector whose components are the expected values of each coordinate.”
  • Meaning: Take expectation of each coordinate separately—expectation acts on vectors entry by entry.

Connected Concepts

• Variance Covariance Matrix
• Normal Distribution in calculus
• Kalman Filter