Expected Value

def. Expected Value. For random variable with countably many outcomes, its expected value is defined as:

Properties. The following identities hold for expected values, with constant , random variables .

  • Linearity
    • If then (reverse does not hold)
  • let be a function over . Then, (Law of Unconscious Statistician)
    • !

thm. Tail Sum Formula. when is a non-negative discrete random variable:

Remark. The Tail Sum formula is useful when the random variable is defined as the minimum or maximum of a certain set of events (e.g. minimum of multiple dice rolls, etc.)

Expectation Manipulation from class:

Conditional Expected Value

def. Conditional Expectation. let be jointly distributed. Then the conditional expected value is defined…

  • …over an event:
  • …over an event on a random variable
  • …over a random variable:
    • ! While expectation conditioned on an event is a value, an expectation conditioned over a random variable is another random variable
    • with more rigour:
    • Intuition. Think of it as “given all information by , what’s the new random variable?”


  • linearity
  • where is a partition of .weighted summation
  • If is -measurable,
  • If is independent of then Taking out independent factors.
  • if is -measurable, Taking out what’s known
    • e.g.
  • Tower Property: if is a random variable, and then:
    • & Think of it as “high-res camera” then “low-res camera”; the final picture is low-res.

thm. Conditional Joint Expectation. and . Then:

thm. Calculating Expected Value from Conditional Expected Value. (Identity 2 above) let be jointly distributed. Then the expected value of is calculated:

  • Useful for computing when depends on .
  • Works regardless of whether are random or discrete, and when mixed.