For the analysis of the birthday paradox, is it important that the birthdays be mutually independent, or is pairwise independence sufficient? Justify your answer.
The key to answering this question is understanding the difference between pairwise independence and mutual independence, and examining where independence is used in the birthday paradox analysis.
Understanding independence levels
Pairwise independence means that any two random variables \(X_i\) and \(X_j\) (\(i \neq j\)) are independent:
\[\Pr\{X_i = a \text{ and } X_j = b\} = \Pr\{X_i = a\} \cdot \Pr\{X_j = b\}\]Mutual independence (also called full independence) means that any subset of the random variables are independent:
\[\Pr\{X_{i_1} = a_1 \text{ and } \cdots \text{ and } X_{i_k} = a_k\} = \Pr\{X_{i_1} = a_1\} \cdots \Pr\{X_{i_k} = a_k\}\]Mutual independence is stronger than pairwise independence.
The birthday paradox analysis
In the birthday paradox, we computed the probability that all \(k\) people have different birthdays. Let \(b_i\) denote person \(i\)’s birthday. We calculated:
\[\begin{align*} \Pr\{\text{all different}\} &= \Pr\{B_k\} \\ &= \Pr\{B_{k-1}\} \cdot \Pr\{A_k \mid B_{k-1}\} \end{align*}\]where \(B_k\) is the event that persons \(1, 2, \ldots, k\) all have different birthdays, and \(A_k\) is the event that person \(k\)’s birthday differs from all previous ones.
The crucial step was computing \(\Pr\{A_k \mid B_{k-1}\}\). Given that the first \(k-1\) people have different birthdays, we needed the probability that person \(k\)’s birthday differs from each of these \(k-1\) specific days.
This requires knowing the joint distribution of \(b_1, b_2, \ldots, b_{k-1}\), not just pairwise relationships. Specifically, we used:
\[\Pr\{b_k \neq b_i \text{ for all } i < k \mid b_1, \ldots, b_{k-1} \text{ all different}\} = \frac{n-(k-1)}{n}\]This calculation assumes that \(b_k\) is independent of the set \(\{b_1, \ldots, b_{k-1}\}\), which requires mutual independence.
Why pairwise independence is not sufficient
Consider a counterexample with \(n = 2\) days and \(k = 3\) people. Suppose we construct birthdays as follows:
- \(b_1 \in \{0, 1\}\) uniformly at random
- \(b_2 \in \{0, 1\}\) uniformly at random
- \(b_3 = b_1 \oplus b_2\) (XOR operation)
This construction gives pairwise independence: any two of \(\{b_1, b_2, b_3\}\) are uniformly distributed and independent. However, they are not mutually independent, because once we know \(b_1\) and \(b_2\), we know \(b_3\) exactly.
With mutual independence, the probability that all three are different is 0 (since there are only 2 days). But our analysis would incorrectly compute:
\[\Pr\{\text{all different}\} = 1 \cdot \frac{1}{2} \cdot \frac{0}{2} = 0\]However, with the XOR construction, if \(b_1 = 0\) and \(b_2 = 1\), then \(b_3 = 1\), giving a collision. The actual probability depends on the specific joint distribution, not just pairwise properties.
Conclusion
Mutual independence is essential for the birthday paradox analysis. Pairwise independence is not sufficient because:
- We need \(b_k\) to be independent of all previous birthdays collectively, not just individually
- The recursive formula \(\Pr\{B_k\} = \Pr\{B_{k-1}\} \cdot \Pr\{A_k \mid B_{k-1}\}\) relies on the joint distribution
- With only pairwise independence, we cannot make valid conclusions about the probability of multiple simultaneous events
The standard birthday paradox assumes each person’s birthday is chosen uniformly and independently from all others, which gives full mutual independence.