Discussion
Let \((X_{1}, X_{2}, X_{3})\) be a random sample from B(1, p). Which one of the following statistics is not sufficient for p ?
Answer: D
The statistic that is not sufficient is \(X_{1} - X_{2} + X_{3}\).
For a Bernoulli sample, the joint probability is \(p^{\sum x_{i}}(1 - p)^{3 - \sum x_{i}}\), so \(T = X_{1} + X_{2} + X_{3}\) is sufficient, and so is any statistic from which T can be computed: \((X_{1}, X_{2} + X_{3})\) and \((X_{1} + X_{2}, X_{3})\) both determine T by adding their components.
\(X_{1} - X_{2} + X_{3}\) does not determine T. The samples (1, 1, 0) and (0, 0, 0) both give the value 0, but their totals are 2 and 0, and their probabilities \(p^{2}(1 - p)\) and \((1 - p)^{3}\) have a ratio that depends on p. So the conditional distribution given this statistic depends on p, and it is not sufficient.
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