Pairwise Independence \neq Mutual Independence

Probability Level 1

Let XX and YY be random variables describing independent tosses of a fair coin. Let ZZ be the random variable that equals 11 if both XX and YY land heads and that equals 0 otherwise.

How many of the following statements are true?

  • The collection of random variables {X,Y}\{X, Y\} is independent.
  • The collection of random variables {X,Z}\{X, Z\} is independent.
  • The collection of random variables {Y,Z}\{Y, Z\} is independent.
  • The collection of random variables {X,Y,Z}\{X, Y, Z\} is independent.
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