Random variables X_{1},X_{2},...,X_{n} are independent and identically distributed. 0 is a parameter of their distribution. If X_{1}, X_{2},...,X_{n}

a2linetagadaW 2021-05-26 Answered
Random variables \(X_{1},X_{2},...,X_{n}\) are independent and identically distributed. 0 is a parameter of their distribution.
If \(X_{1}, X_{2},...,X_{n}\) are Normally distributed with unknown mean 0 and standard deviation 1, then \(\overline{X} \sim N(\frac{0,1}{n})\). Use this result to obtain a pivotal function of X and 0.

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aprovard
Answered 2021-05-27 Author has 20530 answers
Since \(X_{1}, X_{2}, ... ,X_{n}\) are IID random Variables such that,
\(\overline{X}\sim N(0, \frac{1}{n})\)
This implies
\(X_{i}\sim N(0,1)\)
Now,
Pivotal function for Mean can be written as
\(Z=\frac{(\overline{x}-\mu)}{\frac{\sigma}{\sqrt{n}}}=\frac{(\overline{x}-0)}{\frac{1}{\sqrt{n}}}\)
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