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?

Thank you in advance!


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