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I am trying to find what the distribution for $x_1$, $x_2,\dots, x_8$ $\sim$ $N(0,52)$ is. This is a normal distribution question and I'm not adding these probabilities. $x_1$, $x_2,\dots, x_8$ means there are 8 i.i.d variables.

A $N(0, \frac{\bar{x}}{\sqrt{52/8}})$

B $N(0, \frac{1}{\sqrt{52/8}})$

C $N(0,1)$

D $(\bar{x},1)$

E ${\rm Unif}[0,1]$

I am not sure, but I think the answer is A because that has the mean and standard deviation of the sample.

User
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The answer is not a) Since this seems a lot like homework, I’ll just say that you need to consider what distribution does the sum of normally distributed variables follow. After that, it is just simple algebra with the properties of variance.

Sum of normals: https://en.m.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

Vivianne
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  • I am not adding the variables. x1, x2.... x8 is just used to convey that there are 8 variables. Would anything change if I am not adding variables? – User Nov 18 '20 at 20:22
  • @User I don’t think I understand your question then. You have $x1,...,x8$ normally distributed variables and you are looking for the distribution of $ (x1,...,x8)$? If that’s the case, that has PDF equal to the product of the PDFs since they are independent, which by the the way is a Gaussian too: https://math.stackexchange.com/questions/114420/calculate-the-product-of-two-gaussian-pdfs – Vivianne Nov 18 '20 at 22:50