# Likelihood of a random vector with each component following a different distribution

How do you write down the likelihood for random vectors when each component follows a different distribution with a dependence structure?

For example, Suppose there are n-random vectors, mutually independent, $$X_1,\ldots,X_n$$ with 3 components of each vector, i.e. $$X_i=[X_{i,1},X_{i,2},X_{i,3}],i=1,\ldots,n$$. Suppose $$X_{i,1:2}\sim N(\mu,\Sigma)$$ and $$X_{i,3}\sim \text{Exp}(\lambda)$$. Also $$\text{Cov}(X_{i,1},X_{i,3})=a_1$$ and $$\text{Cov}(X_{i,2},X_{i,3})=a_2$$. Then what is the likelihood of this model?

• Are X_{i,1} and X_{i,2} independent? Are X_{i} independent for all i? By the likelihood, do you mean the probability density for this three-variate pdf?
– Alex
Commented Sep 25, 2022 at 8:30
• @Alex X_{i,1} and X_{i,2} need not be independent, $\Sigma$ could have non-zero off-diagonal elements. $X_1,\ldots,X_n$ are mutually independent. Yes, I am looking for the three-variate pdf. Commented Sep 25, 2022 at 8:36
• You cannot derive joint distribution simply by knowing marginal distributions. Commented Sep 25, 2022 at 9:46
• @StubbornAtom There is also information about the covariance between each component. What is the sufficient amount of information needed to obtain the joint distribution? Commented Sep 25, 2022 at 10:00
• Reading up on copulas might give you a place to start. Commented Sep 25, 2022 at 10:17