Studying extreme value r.v. $X=\max_i (c_i+X_i)$ where $c_i$ are constants and $X_i$ are i.i.d. r.v Let
$X_1,X_2,...,X_n$ be independently and identically distributed random variables  according to a distribution $F$.
There are constants: $c_1,c_2,...,c_n$.
Define a new random variable $X=\max_i(X_i+c_i)$.
Is it possible to derive the distribution of $X$ in terms of $F$ and all $c_i$? Based on the derivation, when would $X$ also have the distribution $F$?

In my notation $X_i \sim f$ with CDF $F$. If $X$ is the maximum of $n$ draws, it means that one sample lies at $X$ and the other $n-1$ must lie below it. Thus we have
$$
p(X) = n \cdot f(X) \cdot F(X)^{n-1}
$$
The factors here are i) a combinatiorial factor for the fact that any of the draws could have been the maximum one, ii) a factor for one sample at $X$, iii) a factor for $n - 1$ samples below $X$, written using the CDF.
Now, suppose we want for all $X$
$$
p(X) = f(X) = n \cdot f(X) \cdot F(X)^{n-1}
$$
then we evidently require that for all $X$
$$
n F(X)^{n-1} = 1
$$
which isn't possible enless $n=1$.
