Conditional Expectation Given a Sigma-Algebra · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Forecasting with what you know

Mathematics · Probability · ages 22-23
Name ______________________   Date ____________

  1. The conditional expectation E of X given G must be measurable with respect to G.

    Circle one:   True   False

  2. Roll a fair six-sided die. Given that the roll is odd, what is the conditional expectation of the roll?

    • 4
    • 3.5
    • 3
    • 2

  3. A student says the conditional expectation E[X|G] must be measurable with respect to G. Is that right?

    Circle one:   True   False

  4. Roll a fair six sided die. Given that the roll is even, what is the conditional expectation of the roll?

    • 3
    • 3.5
    • 4

  5. If X is independent of G, a student claims E[X|G] = E[X] almost surely. Is that right?

    Circle one:   True   False

  6. Which identity is the defining property of E[X|G]?

    • E[1_A E[X|G]] = E[1_A X] for every A in G
    • E[1_A E[X|G]] = E[X] for every A in G
    • E[X|G] = E[X] always
    • E[1_A X] = 0 for every A in G

  7. Let X count heads in two fair coin flips, and let G record only the first flip. What is E[ E[X|G] ]?

    Answer: ______________

  8. If X is independent of G, then E of X given G equals E of X almost surely.

    Circle one:   True   False

  9. Two proposed versions of E of X given G disagree on a set of probability one half. What do you conclude?

    • Both are valid, since versions always differ
    • At least one of them is not a genuine version
    • The sigma algebra must be trivial

  10. A student says the tower property E[ E[X|G] |H] = E[X|H] needs X independent of G. Is that right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Forecasting with what you know W1-mt_i2sxZc-ER8-s1

  1. True · Measurability is half the definition: the forecast may only use what G knows.
  2. 3 · Given an odd roll, the live outcomes are 1, 3, 5, whose average is 3.
  3. True · The defining property builds E[X|G] as a G-measurable variable, unique up to a null set.
  4. 4 · The even block holds 2, 4, and 6, whose local mean is 4.
  5. True · Independence makes the extra information useless, so conditioning returns the plain mean.
  6. E[1_A E[X|G]] = E[1_A X] for every A in G · The identity E[1_A E[X|G]] = E[1_A X] for every A in G is the defining property itself.
  7. 1 · The tower property collapses the iterated expectation to E[X] = 1.
  8. True · Independent information teaches nothing, so the forecast collapses to the plain mean.
  9. At least one of them is not a genuine version · Genuine versions differ only on null sets, so a large disagreement disqualifies at least one candidate.
  10. False · The tower property is an identity for nested information with no independence assumed.
Worksheet · LightMySky