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Suppose that X1, ..., Xm are iid Bernoulli(p), Y1, ...., Yn are iid Bernoulli(q), and that...

Suppose that X1, ..., Xm are iid Bernoulli(p), Y1, ...., Yn are iid Bernoulli(q), and that the X's are independent of the Y's where 0 < p < 1 is the unknown parameter with q = 1 - p. By means of the conditional distribution approach, show that sum of Xi - sum of Yi is sufficient for p. {Hint: Instead of looking at the data (X1, ..., Xm, Y1, ...., Yn), can one justify looking at (X1, ..., Xm, 1 - Y1, ...., 1 - Yn)?}

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By factorization theorem we can say that sum xi - sum yi is sufficient stat

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