My question is quite simple. How to prove that a function $f(x_1,x_2,..., x_n)$ is a valid density? I mean, aside of the fact that must integrate to 1, and it must be positive, do i have to prove that all the marginals and conditionals exist or there exists a simple method?
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For Joint Probability Distribution, you don't have to prove anything other than
- integration to 1
- non-negativity
If these conditions hold, your joint PDF is valid. And, all of your marginals also become automatically valid PDFs since: $$\int{f(x_1)dx_1} = \int{\int{f(x_1,x_2,...x_n)dx_n...dx_1}} = 1$$
They'll also be non-negative since the integration of non-negative functions will be non-negative.
