Example 2 Given that $a, b, c, d, e$ are real numbers satisfying $a+b+c+d$ $+e=8$ and $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$, find the range of values for $e$. (Example 1 from [1])
By Cauchy-Schwarz inequality, we have $\left(1^{2}+1^{2}+1^{2}+1^{2}\right)\left(a^{2}+b^{2}+c^{2}+d^{2}\right) \geqslant(a+b+c+d)^{2}, \therefore 4\left(16-e^{2}\right) \geqslant(8-e)^{2}, \therefore 0 \leqslant e \leqslant \frac{16}{5}$.

Using the same approach as in Example 2 of this article, we can solve Example 2 and Example 3 in [1], which are not detailed here. That is,

Let $n \in \mathbf{N}, a, b \in \mathbf{R}, x_{0}, x_{1}, x_{2}, \cdots, x_{n}$ be real variables, and $\sum_{i=0}^{n} x_{i}=a, \sum_{i=0}^{n} x_{i}^{2}=b$, find the range of $x_{0}$ (Example 2 in [1]);

Given $n \in \mathbf{N}_{+}, x_{i} \in \mathbf{R}, i=1,2, \cdots, n$, and $\sum_{i=0}^{n} x_{i}=a$, find the range of $\sum_{i=0}^{n} x_{i}^{2}$ (Example 3 in [1]).