# 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}

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
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
$$Z=\frac{(\overline{x}-\mu)}{\frac{\sigma}{\sqrt{n}}}=\frac{(\overline{x}-0)}{\frac{1}{\sqrt{n}}}$$