Pairwise Independence \(\neq\) Mutual Independence

Let \(X\) and \(Y\) be random variables describing independent tosses of a fair coin. Let \(Z\) be the random variable that equals \(1\) if both \(X\) and \(Y\) land heads and that equals 0 otherwise.

How many of the following statements are true?

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