Let $X_1, X_2 \ldots $ be i.i.d. r.v.s with CDF $F$, and let $M_n = \max(X_1, X_2,...,X_n)$. Find the joint distribution of $M_n$ and $M_{n+1}$, for each $n\ge1$.
My solution is the following:
The probability $P(M_n \le t, M_{n+1} \le k)$ has two cases. Either $k \lt t$ or $k \ge t$.
In case $k \lt t$: $$P(M_n \le t, M_{n+1} \le k) = P(M_n \le t| M_{m+1} \le k)P(M_{n+1} \le k) = $$ $$P(X_1 \le t, X_2 \le t, \ldots X \le t|X_1 \le k, \ldots X_{n+1} \le k)F(k)^{n+1} = F(k)^{n+1}$$
In case $k \ge t$: $$P(M_n \le t, M_{n+1} \le k) = P(M_n \le t| M_{m+1} \le k)P(M_{n+1} \le k) = $$ $$P(X_1 \le t, X_2 \le t, \ldots X \le t|X_1 \le k, \ldots X_{n+1} \le k)F(k)^{n+1} = $$ $$P(X_1 \le t | X_1 \le k) \ldots P(X_n \le t | X_n \le k)F(k)^{n+1} = F(t)^nF(k)$$
This means: $$P(M_n \le t, M_{n+1} \le k) = \left\{ \begin{array}{ll} F(k)^{n+1} & k \lt t \\ F(t)^nF(k) & k \ge t\ \ \end{array} \right. $$
I'd appreciate if somebody could check if my reasoning is correct.