解答
1
$$\mathbb{F}_{X_h}(x)\equiv\mathbb{P}\left(X_h\leq x\right) = \int_0^x1dx^{\prime} = x$$
2
$$\begin{aligned}
\mathbb{F}_{X_{\max}}(x)
&\equiv\mathbb{P}\left(X_{\max}\leq x\right) = \mathbb{P}\left(X_{(1)}\leq x,X_{(2)}\leq x,\ldots,X_{(n)}\leq x\right)\\
&=\int_0^x1dx_{(1)}^{\prime}\int_0^x1dx_{(2)}^{\prime}\cdots\int_0^x1dx_{(n)}^{\prime} = x^n
\end{aligned}$$
3
$$f_{X_{\max}}(x) = \frac{d\mathbb{F}_{X_{\max}}}{dx}(x) = \frac{d}{dx}x^n = nx^{n-1}$$
4
$$\mathbb{E}\left(X_{\max}\right) = \int_0^1xf_{X_{\max}}(x)dx = \int_0^1nx^ndx = \frac{n}{n+1}\left[x^{n+1}\right]_0^1 = \frac{n}{n+1}$$
5
$$\begin{aligned}
\mathbb{F}_{X_{\min}}(x)
&\equiv\mathbb{P}\left(X_{\min}\leq x\right) = 1 - \mathbb{P}\left(X_{(1)}> x,X_{(2)}> x,\ldots,X_{(n)}> x\right)\\
&=1 - \int_x^11dx_{(1)}^{\prime}\int_x^11dx_{(2)}^{\prime}\cdots\int_x^11dx_{(n)}^{\prime} = 1 - \left(1-x\right)^n
\end{aligned}$$