Distribution of maxima of N independent but non-identical exponential family distributions?

Question

For an N-dimensional i.i.d. random vector $X=(X_1,...,X_N)$, with (identical) component-wise c.d.f.s $F(x_i)$, the distribution of the maximum is $F(x_i)^N$. But I am interested in the maxima distribution for the case of mutually independent but non-identical random variables. Either generally (if this is known), for the case of the exponential family, or the Gaussian case. Thanks.

Answer 1

If the $X_i$'s are independent, the general formula is $$ F_{\max_{i=1,\ldots,N} X_i}(t)=P(\max_{i=1,\ldots,N} X_i\le t) = P(X_1\le t \cap\cdots \cap X_N\le t) = \prod_{i=1}^N P(X_i\le t)= \prod_{i=1}^N F_{X_i}(t). $$ If $X_i$ has the exponential distribution with rate $\lambda_i$, then substituting $$ F_{\max_{i=1,\ldots,N} X_i}(t) = \prod_{i=1}^N (1-e^{-\lambda_i t}). $$