How to specify joint distribution when they are not jointly independent?

Question

I have three variables, $X, Y, Z$ with marginal distributions $F_x, F_y, F_z$.

I want to specify the joint distribution $F_{xyz}$ of $(X, Y, Z)$.

I know that if $X, Y, Z$ are jointly independent, then $$F_{xyz} = F_x F_y F_z.$$

However, what if $(Y,Z)$ are dependent with some joint distribution $F_{yz}$, but $X$ is jointly independent from $(Y, Z)$.

How do I then specify the joint distribution? Is it simply $$F_{xyz} = F_xF_{yz}?$$

Is such a distribution guaranteed to exist?

gunes · Answer 1 · 2019-01-06T13:18:47.040

1

By definition, we can always write $p(x,y,z)=p(x|y,z)p(y,z)$ by definition.When $X$ is independent from $(Y,Z)$ pair, $p(x|y,z)$ becomes $p(x)$. So, joint distribution can be written as you suggest, i.e. $F_{xyz}=F_xF_{yz}$. By the way, I said $X$ is independent from $(Y,Z)$ because when you have only two elements, independence and joint independence are equivalent since there are no further subsets.

edited Jan 06 '19 at 13:18

answered Jan 05 '19 at 19:12

gunes

57,205

p(x,y,z)=p(x|y,z)p(y,z) is just conditional probability rather than Bayes. Bayes involves interchanging the variables on either side of the conditioning – Glen_b Jan 06 '19 at 13:16
It's been like a tradition for me to associate any simple conditional probability formula with Bayes Law :) – gunes Jan 06 '19 at 13:19

How to specify joint distribution when they are not jointly independent?

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