Joint Distributions, Covariance and Independence · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Two variables moving together

Mathematics · Probability · ages 19-20
Name ______________________   Date ____________

  1. A joint table gives P(X=0,Y=0)=0.4, P(X=0,Y=2)=0.1, P(X=1,Y=0)=0.2, P(X=1,Y=2)=0.3. Find the marginal chance P(X=1).

    Answer: ______________

  2. A joint table gives P(X=0,Y=0)=0.4, P(X=0,Y=2)=0.1, P(X=1,Y=0)=0.2, P(X=1,Y=2)=0.3. Find P(X=1).

    Answer: ______________

  3. The covariance of X and Y is positive. What does that sign tell you?

    • X causes Y to change
    • Above average X tends to come with above average Y
    • X and Y are independent

  4. Using the same joint table, find P(Y=2).

    Answer: ______________

  5. Lee says a covariance of zero proves X and Y are independent. Is Lee right?

    Circle one:   True   False

  6. The covariance of X and Y is positive. What does that sign tell you?

    • Above average X tends to come with above average Y
    • X causes Y to change
    • X and Y are independent
    • The link between X and Y is strong

  7. Lee says a covariance of zero proves X and Y are independent.

    Circle one:   True   False

  8. For the same table, E[X] = 0.5 and E[Y] = 0.8. Compute Cov(X, Y).

    Answer: ______________

  9. For the same table, E[X] = 0.5 and E[Y] = 0.8. Compute Cov(X, Y).

    Answer: ______________

  10. X is symmetric about 0 and Y = X squared, so Cov(X, Y) = 0. What follows?

    • X and Y must be independent
    • The covariance computation must be wrong
    • They stay dependent: Y is fixed by X
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Answer key

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

Two variables moving together W1-mt_YyTFTAqhSF-s1

  1. 0.5 · Add across the X = 1 column: 0.2 + 0.3 = 0.5.
  2. 0.5 · Add across the X = 1 column: 0.2 + 0.3 = 0.5.
  3. Above average X tends to come with above average Y · The sign reports co-movement direction, nothing about causes.
  4. 0.4 · Add down the Y = 2 row: 0.1 + 0.3 = 0.4.
  5. False · False. Take X symmetric around zero with Y = X squared: covariance is zero by symmetry, yet Y is fully set by X.
  6. Above average X tends to come with above average Y · The sign only reports co-movement direction: positive means above average values tend to appear together. It says nothing about causes or strength.
  7. False · Symmetric X with Y = X squared gives zero covariance with total dependence.
  8. 0.2 · E[XY] = 0.6 minus E[X]E[Y] = 0.4 leaves 0.2.
  9. 0.2 · E[XY] = 2 x 0.3 = 0.6, while E[X] x E[Y] = 0.5 x 0.8 = 0.4. The covariance is 0.6 - 0.4 = 0.2.
  10. They stay dependent: Y is fixed by X · Symmetry zeroes the cross term while Y remains fully set by X.
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