# Is linear combination of dependent exponential family random variables(multinomial, normal) in exponential family?

As well known to us, multinomial distribution and normal distribution belong to exponential family. Define $$Z = \sum_{i=1}^{n}a_{i}(X_{i} - b_{i}) + \sum_{j=1}^{m}c_{j}Y_{j},$$ where $$X_{i}$$ is multinomial random variable, $$Y_{j}$$ is normal random variable, $$a_{i}, b_{i}, c_{j}$$ are constants. Additionally, $$X_{i}, Y_{j} (i = 1,2,\cdots,n, j = 1,2,\cdots,m)$$ are dependent.

Question: does the distribution of random variable $$Z$$ belong to exponential family?

Furthermore, in general, does the distribution of linear combination of exponential family random variables belong to exponential family?

If yes, can you provide a rigorous proof or a reference? Otherwise, can you provide a counterexample?