Find distribution and expectation of $Z_1=\max(1,X_1,X_2)$ and $Z_2=\min(2,X_1,X_2)$ where $X_1$ and $X_2$ have distribution function $$F(x) = 1 - e^{-x}.\quad x\ge0$$
My thoughts: For $Z_1 = \min(2,X_1,X_2)$, \begin{align*} F_{Z_1}(x) &= P(Z_1 \le x) = 1 - P(Z_1 > x) = 1 - P(\min(2, X_1,X_2)\gt x)\\ &= 1 - P(2 > x)P(X_1>x)P(X_2>x)\\ &= \begin{cases}1 - e^{-2x}, 0<x\le2\\0, x>2\end{cases} \end{align*} and $E[Z_1] = \int_{-\infty}^{\infty}x·2e^{-x}\,\mathrm dx$ as I understand. For $Z_2 = \max(1,X_1,X_2)$, \begin{align*} F_{Z_2}(x) &= P(Z_2 \le x) = P(\max(1, X_1,X_2)\le x)\\ &= P(1 \le x)P(X_1\le x)P(X_2\le x)\\ &= \begin{cases}1 - e^{-2x}, x>1\\0, 0<x\le1\end{cases} \end{align*} and $E[Z_2] = \int_{-\infty}^{\infty}x·2e^{-x}\,\mathrm dx$.
Am I right with my assumtions?