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If $x_i \sim \mathrm{N}(0,1)$ for i = 1:n and $x_i$ are iid, is it true that $(x_1, x_2, ...,x_{n}) \sim \mathrm{N}(0,I)$ where I is identity matrix of size n?

If $x_i \sim \mathrm{N}(0,\sigma)$ for i = 1:n and $x_i$ are iid, is it true that $(x_1, x_2, ...,x_{n}) \sim \mathrm{N}(0,\sigma I)$ where I is identity matrix of size n?

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Yes.

You can show this through many ways. I think the easiest way would be to check if the product of their moment generating functions is equal to the multivariate normal moment generating function.

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