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# please help me with probability HW, i will rate the best answer, thank you very much....

please help me with probability HW, i will rate the best answer, thank you very much. if you're not sure with your answer please do me a favor and let someone else do it cus i can only post the question once. i appreciate the help !

a general two-sided 100(1 - a)% confidence interval for the mean /i when o is known: where a1a2 = a. Show that the length of the interval, o(za, + za,)//n is minimized when a1 = a2 = a/2.

Let us denote

To find which confidence interval is shorter, suppose is a confidence interval. Then, the length of the interval is .

The minimization problem is
minimize
Subject to
Using Lagrange multiplier, the conditions are,

Now, using Fundamental Theorem of calculus,

Substituting in equations (1) and (2),

Adding equations (3) and (4), we get

does not make any sense. So

Thus the interval is shorter than

Here
Proof is complete.

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