Question

# Consider the family of distributions with pmf pX(x) = p if x = −1, 2p if...

Consider the family of distributions with pmf pX(x) = p if x = −1, 2p if x = 0, 1 − 3p if x = 1 . Here p is an unknown parameter, and 0 ≤ p ≤ 1/3. Let X1, X2, . . . , Xn be iid with common pmf a member of this family. Consider the statistics A = the number of i with Xi = −1, B = the number of i with Xi = 0, C = the number of i with Xi = 1.

Write down the joint pmf of X1, X2, · · · , Xn.

Solution

Since X1, X2, . . . , Xn are iid, their joint probability = product of their individual probabilities.

There are A Xi’s whose value is – 1 for which probability is p. Hence, joint probability = pA.

Similarly, for B Xi’s whose value is 0 for which probability is 2p, joint probability = (2p)B.

and for C Xi’s whose value is 0 for which probability is (1 – 3p), joint probability = (1 - 3p)C.

Thus, joint probability pmf of X1, X2, . . . , Xn is:

(2B(pA+B)(1 - 3p)C, A + B + C = n and 0 ≤ p ≤ 1/3. Answer

DONE

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