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1

First, a discrete distribution (or a distribution with atoms) cannot be transformed such to a multinormal, so assume an absolutely continuous distribution. Then, if $p=1$, we can always transform to normal. So assume $p>1$. Then, if the components are independent, again, we can use the univariate solution. So there must be some dependencies between the ...

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I think the following example could do it : Let $X_1 \sim \mathcal{N}(0, 1)$ and $B$ be a binary variable which is equal to $1$ with probability $0.5$ and $-1$ with probability $0.5$. Define $X_2 = B\times X_1$. Then the marginal distributions of $X_1$ and $X_2$ are obviously $\mathcal{N}(0, 1)$ but $(X_1; X_2)$ is not a multivariate Gaussian. An easy ...

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The usual setting for estimating a joint pdf, is of having paired observations, say a sample of the form $(x_1, y_1), (x_2, y_2), \dotsc, (x_n,y_n)$. Then we can do business. If you have less than that, you should tell us more about your context, and why ... There are many possibilities. If you know your two variables are independent, then the marginals ...

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