Maybe this is too basic but i must seek some assistance and confirmation here. Assume a function F with f(x_1, x_2,..., x_n)1) F is linear and its variables are distributed normally. Is F distributed normally too?2) F is linear and some of its variables are distributed normally and some lognormally. Is F distributed normally too?3)F is non linear and its variables are distributed normally. Is F distributed normally too?4) F is non linear and its variables are distributed some normally and some lognormally. Is F distributed normally too?If the answer to any of the above is No, please explain the reason why. Is there any other parametric type of statistical distribution that should fit in this setting?Thanks in advance.

1) is true if the variables are _jointly_ normal.All the others are false in general, although they may in some special cases be true in either an exact or an approximate sense.For instance, if the variables are independent and the coefficients are similar in size then 2) is true for large n by the central limit theorem.

Thanks pal. You have just confirmed my suspicions.

In what context are you using this ?

I agree with MrMartingale's reply. In fact, the definition of jointly Normal (or Multivariate Normal) is that all linear combinations of the marginal distributions are Normal.If x_1 and x_2 are both Normal(0,1) but not bivariate Normal then you might have something like x_2 = x_1 for |x_1| < 1 and x_2 = -x_1 for |x_1|>=1. The linear combination x_1 + x_2 has a point mass at 0 with probability 0.3173 plus a N(0,2) distribution truncated at -2 and 2. That is certainly not Normal.(2) is approximately true for any finite variance marginal distributions, if you make n big and keep the linear weights roughly even in size. That's the Central Limit Theorem.(3) and (4) are basically not true. You'd have to construct some degenerate case (like Normal distributions with zero standard deviation) to make them true. They're certainly not true in general.