2015

Problem 1

Let $\mathbf{A}$ and $\mathbf{b}$ be defined as

$$\mathbf{A}=\left(\begin{array}{ccc}-3& 0& 0\\ -2& -3& 1\\ 2& -3& -3\end{array}\right),\mathbf{b}=\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right).$$

The partial derivative of a scalar-valued function $f(x)$ with respect to $\mathbf{x}=(x_1{x}_2{x}_3{)}^T$ is defined as

$$\frac{\partial }{\partial \mathbf{x}}f(x)=\left(\frac{\partial }{\partial x_1}f(x)\frac{\partial }{\partial x_2}f(x)\frac{\partial }{\partial x_3}f(x)\right),$$

and a stationary point of $f(x)$ is defined as $\mathbf{x}$ satisfying $\frac{\partial }{\partial \mathbf{x}}f(x)=(000)$. $x^T$ denotes the transpose of $\mathbf{x}$. Answer the following questions.

1. Find the characteristic polynomial of $\mathbf{A}$.
2. $\mathbf{C}$ is given as $\mathbf{C}={\mathbf{A}}^5+9{\mathbf{A}}^4+30{\mathbf{A}}^3+36{\mathbf{A}}^2+30\mathbf{A}+9\mathbf{I}$ by using $\mathbf{A}$ and an identity matrix $\mathbf{I}$. Calculate $\mathbf{C}$.
3. Calculate the partial derivative of $x^T\mathbf{A}x$ with respect to $\mathbf{x}$.
4. Find a symmetric matrix $\tilde{\mathbf{A}}$ that satisfies the equation $x^T\mathbf{A}x=x^T\tilde{\mathbf{A}}x$ for any vector $\mathbf{x}$. Find eigenvalues ${\lambda }_1,{\lambda }_2,{\lambda }_3$ $({\lambda }_1\ge {\lambda }_2\ge {\lambda }_3)$, and eigenvectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$. Choose the eigenvectors such that $\mathbf{V}=({\mathbf{v}}_1{\mathbf{v}}_2{\mathbf{v}}_3)$ becomes an orthogonal matrix.
5. Prove that $x^T\mathbf{A}x\le 0$ holds for any real vector $\mathbf{x}$.
6. Find a stationary point of function $g(x)=x^T\mathbf{A}x+2{\mathbf{b}}^T\mathbf{x}$.

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以下是问题 1

令 $\mathbf{A}$ 和 $\mathbf{b}$ 定义为

$$\mathbf{A}=\left(\begin{array}{ccc}-3& 0& 0\\ -2& -3& 1\\ 2& -3& -3\end{array}\right),\mathbf{b}=\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right).$$

标量值函数 $f(x)$ 相对于 $\mathbf{x}=(x_1{x}_2{x}_3{)}^T$ 的偏导数定义为

$$\frac{\partial }{\partial \mathbf{x}}f(x)=\left(\frac{\partial }{\partial x_1}f(x)\frac{\partial }{\partial x_2}f(x)\frac{\partial }{\partial x_3}f(x)\right),$$

$f(x)$ 的驻点定义为满足 $\frac{\partial }{\partial \mathbf{x}}f(x)=(000)$ 的 $\mathbf{x}$。$x^T$ 表示 $\mathbf{x}$ 的转置。回答以下问题。

1. 求 $\mathbf{A}$ 的特征多项式。
2. 给定 $\mathbf{C}={\mathbf{A}}^5+9{\mathbf{A}}^4+30{\mathbf{A}}^3+36{\mathbf{A}}^2+30\mathbf{A}+9\mathbf{I}$，使用 $\mathbf{A}$ 和单位矩阵 $\mathbf{I}$ 计算 $\mathbf{C}$。
3. 计算 $x^T\mathbf{A}x$ 对 $\mathbf{x}$ 的偏导数。
4. 找到一个对称矩阵 $\tilde{\mathbf{A}}$，使得对于任意向量 $\mathbf{x}$，方程 $x^T\mathbf{A}x=x^T\tilde{\mathbf{A}}x$ 成立。求特征值 ${\lambda }_1,{\lambda }_2,{\lambda }_3$ $({\lambda }_1\ge {\lambda }_2\ge {\lambda }_3)$ 和特征向量 ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$。选择特征向量使得 $\mathbf{V}=({\mathbf{v}}_1{\mathbf{v}}_2{\mathbf{v}}_3)$ 成为正交矩阵。
5. 证明 $x^T\mathbf{A}x\le 0$ 对于任意实向量 $\mathbf{x}$ 成立。
6. 求函数 $g(x)=x^T\mathbf{A}x+2{\mathbf{b}}^T\mathbf{x}$ 的驻点。

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Problem 2

Answer the following questions regarding curves on the $xy$-plane.

1. Show that the foci of an ellipse:

   $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\text{(*)}$$

   and those of a hyperbola:

   $$\frac{x^2}{c^2}-\frac{y^2}{d^2}=1(c>d>0)\text{(**)}$$

   are $\left(\pm \sqrt{a^2-b^2},0\right)$ and $\left(\pm \sqrt{c^2+d^2},0\right)$, respectively. Note that an ellipse (hyperbola) is a curve such that the sum (difference) of the distances from the foci to any point on the curve is constant.

2. As for Eq. (*), consider the set $E_u$ of ellipses such that $a^2-b^2=u^2$ ($u$ is a positive constant). By writing the simultaneous equations that consist of Eq. (*) and the differential equation obtained by taking the derivative of Eq. (*) with respect to $x$, show that any ellipse in $E_u$ satisfies

   $$xy(y^{\prime }{)}^2+\left(x^2-y^2-u^2\right)y^{\prime }-xy=0,\text{(***)}$$

   where $y^{\prime }=\frac{\mathrm{d}y}{\mathrm{d}x}$.

3. As for Eq. (**), consider the set $H_u$ of hyperbolae such that $c^2+d^2=u^2$. Show that any hyperbola in $H_u$ satisfies Eq. (***).

4. Let $C_u$ be the set of curves perpendicular to any ellipse in $E_u$. Let $D_u$ be the set of curves obtained by removing from $C_u$ the line $x=0$ as well as all the curves including a point such that $y^{\prime }=0$. Find a differential equation that any curve in $D_u$ satisfies.

5. Solve the differential equation that you found in Question (4). If necessary, rewrite the differential equation into a differential equation with respect to $p$ with replacement such that $\alpha =x^2$, $\beta =y^2$, and $p=\frac{\mathrm{d}\beta }{\mathrm{d}\alpha }$.

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在 $xy$ 平面上回答以下与曲线有关的问题。

1. 证明椭圆的焦点：

   $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\text{(*)}$$

   和双曲线的焦点：

   $$\frac{x^2}{c^2}-\frac{y^2}{d^2}=1(c>d>0)\text{(**)}$$

   分别是 $\left(\pm \sqrt{a^2-b^2},0\right)$ 和 $\left(\pm \sqrt{c^2+d^2},0\right)$。注意，椭圆（双曲线）是一种曲线，使得从焦点到曲线上任意一点的距离之和（差）是恒定的。

2. 对于方程 (*)，考虑椭圆 $E_u$ 的集合，使得 $a^2-b^2=u^2$ （$u$ 是一个正常数）。通过写出由方程 (*) 及其相对于 $x$ 的导数方程组成的联立方程，证明 $E_u$ 中的任何椭圆都满足

   $$xy(y^{\prime }{)}^2+\left(x^2-y^2-u^2\right)y^{\prime }-xy=0,\text{(***)}$$

   其中 $y^{\prime }=\frac{\mathrm{d}y}{\mathrm{d}x}$。

3. 对于方程 (**) ，考虑双曲线 $H_u$ 的集合，使得 $c^2+d^2=u^2$。证明 $H_u$ 中的任何双曲线都满足方程 (***)。

4. 设 $C_u$ 为与 $E_u$ 中任何椭圆垂直的曲线集合。设 $D_u$ 为通过去除 $C_u$ 中直线 $x=0$ 以及包含 $y^{\prime }=0$ 点的所有曲线而获得的曲线集合。找出 $D_u$ 中任何曲线满足的微分方程。

5. 求解你在问题 (4) 中找到的微分方程。如果有必要，用 $p$ 代替使得 $\alpha =x^2$, $\beta =y^2$，并且 $p=\frac{\mathrm{d}\beta }{\mathrm{d}\alpha }$ 的微分方程。

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Problem 3

Answer the following questions.

1. Let $X$ be a real-valued random variable. Let $t$ be a real-valued variable. We define ${\varphi }_X(t)$ for $X$ as

   $${\varphi }_X(t)=E_X\left[e^{tX}\right],$$

   where $E_X[\cdot ]$ denotes the expectation taken with respect to $X$. Supposing that ${\varphi }_X(t)$ is finite in a neighborhood of $t=0$, give the mean and variance of $X$ using ${\varphi }_X^{\prime }(0)$ and ${\varphi }_X^{\prime \prime }(0)$. Here ${\varphi }_X^{\prime }(t)$ and ${\varphi }_X^{\prime \prime }(t)$ denote the first- and second-order derivatives of ${\varphi }_X(t)$ with respect to $t$, respectively.

2. For a sequence of mutually independent random variables: $X_1,X_2,\dots ,X_N$, suppose that each $X_j$ is identically generated according to the 1-dimensional normal distribution with mean $\mu$ and variance ${\sigma }^2$. That is, the probability density function for each $X_j$ is given by

   $$p(X_j=x)=\frac{1}{\sqrt{2\pi \sigma }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right).$$

   Then calculate ${\varphi }_{X_j}(t)$. Also find a probability distribution according to which

   $$Y=X_1+X_2+\cdots +X_N$$

   is generated. You can use the fact that for random variables $Z$ and $W$ with ${\varphi }_Z(t)={\varphi }_W(t)$, the probability distribution of $Z$ is the same as that of $W$.

3. Suppose that $N\in \{1,2,\dots ,\infty \}$ as in Question (2) is generated according to the geometric distribution with parameter $\theta$ $(0<\theta <1)$ for which the probability function is given by

   $$P(N=n)=(1-\theta )n\theta .$$

   For $Y=X_1+X_2+\cdots +X_N$, define ${\varphi }_Y(t)$ by

   $${\varphi }_Y(t)=E_Y\left[e^{tY}\right].$$

   Then calculate ${\varphi }_Y(t)$ and express it using ${\varphi }_{X_j}(t)$. Since ${\varphi }_{X_j}(t)$ does not depend on $j$, you can write it as ${\varphi }_X(t)$.

4. Calculate the mean and variance of $Y$ in Question (3).

5. Given $\xi (>E_Y[Y])$, give an upper bound on the probability that $Y$ in Question (3) exceeds $\xi$, as a function of $\mu$, $\sigma$, $\theta$, and $\xi$ (not all of $\mu$, $\sigma$, $\theta$, and $\xi$ have to be used).

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回答以下问题。

1. 设 $X$ 是一个实值随机变量。令 $t$ 是一个实值变量。我们定义 $X$ 的 ${\varphi }_X(t)$ 为

   $${\varphi }_X(t)=E_X\left[e^{tX}\right],$$

   其中 $E_X[\cdot ]$ 表示对 $X$ 取期望。假设 ${\varphi }_X(t)$ 在 $t=0$ 的邻域内是有限的，使用 ${\varphi }_X^{\prime }(0)$ 和 ${\varphi }_X^{\prime \prime }(0)$ 给出 $X$ 的均值和方差。这里 ${\varphi }_X^{\prime }(t)$ 和 ${\varphi }_X^{\prime \prime }(t)$ 分别表示 ${\varphi }_X(t)$ 的一阶和二阶导数。

2. 对于一列相互独立的随机变量：$X_1,X_2,\dots ,X_N$，假设每个 $X_j$ 都按一维正态分布生成，均值为 $\mu$，方差为 ${\sigma }^2$。即每个 $X_j$ 的概率密度函数为

   $$p(X_j=x)=\frac{1}{\sqrt{2\pi \sigma }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right).$$

   然后计算 ${\varphi }_{X_j}(t)$。还要找到一个概率分布，使得

   $$Y=X_1+X_2+\cdots +X_N$$

   是生成的。你可以使用这样一个事实，对于随机变量 $Z$ 和 $W$，如果 ${\varphi }_Z(t)={\varphi }_W(t)$，则 $Z$ 的概率分布与 $W$ 相同。

3. 假设 $N\in \{1,2,\dots ,\infty \}$（如问题 (2)）是按照参数为 $\theta$ $(0<\theta <1)$ 的几何分布生成的，其概率函数为

   $$P(N=n)=(1-\theta )n\theta 。$$

   对于 $Y=X_1+X_2+\cdots +X_N$，定义 ${\varphi }_Y(t)$ 为

   $${\varphi }_Y(t)=E_Y\left[e^{tY}\right]。$$

   然后计算 ${\varphi }_Y(t)$ 并用 ${\varphi }_{X_j}(t)$ 表示它。由于 ${\varphi }_{X_j}(t)$ 不依赖于 $j$，你可以将其写为 ${\varphi }_X(t)$。

4. 计算问题 (3) 中 $Y$ 的均值和方差。

5. 给定 $\xi (>E_Y[Y])$，给出问题 (3) 中 $Y$ 超过 $\xi$ 的概率的上界，作为 $\mu$、$\sigma$、$\theta$ 和 $\xi$ 的函数（不必全部使用 $\mu$、$\sigma$、$\theta$ 和 $\xi$）。