If $X_1,\ldots,X_i,\ldots,X_n$ are same normal distribution, $X_i \sim \operatorname{Normal}(0,σ^2)$,
and they are independent.
$$
Z = \frac{\sum_{i=1}^n X_i^2} n.
$$
What is the distribution of the square of the normal distribution?like $X_i^2$,and,what is it mean and variance?
I am trying to turn this Z into a normal distribution
can we use chi-square distribution and central limit theorem to find the approximate normal distribution ?
How to do it?
I do not quite understand the chi-square distribution and central limit theorem,
could you answer this question in detail?
Any help would be much appreciated!
re-edit:
I do this works:
$$
Z = \frac{\sum_{i=1}^n X_i^2} n= σ^2\sum_{i=1}^n \left(\frac{X_i}{σ}\right)^2.
$$
this is a chi-square distribution,and mean $= nσ^2$, var${}=2nσ^2$.
is this right?
and how to use CLT to find the approximate normal distribution?
Best Answer
There is no positive normal random variable!