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

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?

## 1 Answer

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.

• 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 6 at 13:16
• It's been like a tradition for me to associate any simple conditional probability formula with Bayes Law :) – gunes Jan 6 at 13:19