Discussion
Suppose that \(\Omega_{1} = \{1, 2, 3, 4, 5, 6\}\) and that P is a probability measure such that \(P(\{j\}) = 1/6\), for \(j = 1, 2, 3, 4, 5, 6\). Let \(I_{A}\) and \(I_{B}\) be the indicator functions of the subsets \(A = \{1, 3, 5\}\) and \(B = \{2, 3, 4, 6\}\) respectively. The following are two statements :
(I) \(I_{A}\) and \(I_{B}\) are identically distributed
(II) \(I_{A}\) and \(I_{B}\) are independent random variables
Which of the following is true ?
Answer: C
Neither (I) nor (II) is true.
\(I_{A}\) is 1 with probability \(P(A) = 3/6 = 1/2\), while \(I_{B}\) is 1 with probability \(P(B) = 4/6 = 2/3\). Two indicator variables are identically distributed only when the events have equal probability, so (I) fails.
For independence, compare \(P(A \cap B)\) with \(P(A)P(B)\):
\[ \begin{aligned} P(A \cap B) &= P(\{3\}) = \frac{1}{6} \\ P(A)P(B) &= \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \end{aligned} \]
These differ, so A and B, and hence \(I_{A}\) and \(I_{B}\), are not independent. (II) fails too, which rules out every option asserting either statement.
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