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I need to express a univariate normal as a multivariate normal to make certain calculations possible (for example: being able to divide two Gaussian distributions). So, my univariate normal is defined over only one random variable ex: $G(x)$ and now I want to define it over $G(x_1, x_2.....x_n)$. However, even though it is defined over $n$ random variables, it does not depend on the values of the other random variables.

Can someone tell me how I can define the mean and covariance of such a distribution, so it is mathematically correct? Does the mean vector only have non-zero value at one location or should I try and describe $(x-\mu)$ as 0? What about the covariance matrix?

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The mean vector should all have the same mean ($\mu$) and the covariance matrix is the identity matrix scaled by the variance of the univariate normal ($\sigma^2I$).

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