3.4 Expected Value

For a continuous random variable \(X\) with PDF \(f\), the expected value of \(X\), written \(E[X]\) is

\[ E[X] = \int_{-\infty}^{\infty}xf_{X}(x) dx \]

Oh, ok. If you say so. (We do…).

There are two really important things to grasp here:

  1. What does this mean about a particular PDF?
  2. What is the justification for this particualar definition?

With your instructor, talk about what each of the following definitions mean in your own words. For key concepts, you might also formalize this intuition into a formula that can be computed.

  • Expected Value, or Expectation
  • Central Moments \(\rightarrow\) Variance \(\rightarrow\) Standard Deviation
  • Set aside for later: Chebyshev’s Inequality and the Normal Distribution
  • Mean Squared Error and its alternative formula
  • Covariance and Correlation