Let $A$ and $b$ be defined as
$$A = \left( \begin{array} { r r r } - 3 & 0 & 0 \\ - 2 & - 3 & 1 \\ 2 & - 3 & - 3 \end{array} \right) , b = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) .$$
The partial derivative of a scalar-valued function $f ( x )$ with respect to $x = \left( \begin{array} { l l l } x _ { 1 } & x _ { 2 } & x _ { 3 } \end{array} \right) ^ { T }$ is defined as
$$\frac { \partial } { \partial x } f ( x ) = \left( \frac { \partial } { \partial x _ { 1 } } f ( x ) \quad \frac { \partial } { \partial x _ { 2 } } f ( x ) \quad \frac { \partial } { \partial x _ { 3 } } f ( x ) \right)$$
and a stationary point of $f ( x )$ is defined as $x$ satisfying $\frac { \partial } { \partial x } f ( x ) = \left( \begin{array} { l l l } 0 & 0 & 0 \end{array} \right) . x ^ { T }$ denotes the transpose of $x$. Answer the following questions.
(1) Find the characteristic polynomial of $A$.
(2) $C$ is given as $C = A ^ { 5 } + 9 A ^ { 4 } + 30 A ^ { 3 } + 36 A ^ { 2 } + 30 A + 9 I$ by using $A$ and an identity matrix $I$. Calculate $C$.
(3) Calculate the partial derivative of $x ^ { T } A x$ with respect to $x$.
(4) Find a symmetric matrix $\tilde { A }$ that satisfies equation $x ^ { T } A x = x ^ { T } \tilde { A } x$ for any vector $x$. Find eigenvalues $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \left( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \lambda _ { 3 } \right)$, and eigenvectors $v _ { 1 } , v _ { 2 } , v _ { 3 }$. Choose the eigenvectors such that $V = \left( v _ { 1 } v _ { 2 } v _ { 3 } \right)$ becomes an orthogonal matrix.
(5) Prove that $x ^ { T } A x \leq 0$ holds for any real vector $x$.
(6) Find a stationary point of function $g ( x ) = x ^ { T } A x + 2 b ^ { T } x$.

Answer the following questions regarding curves on the $x y$-plane.
(1) Show that the foci of an ellipse:
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \quad ( a > b > 0 )$$
and those of a hyperbola:
$$\frac { x ^ { 2 } } { c ^ { 2 } } - \frac { y ^ { 2 } } { d ^ { 2 } } = 1 \quad ( c > d > 0 )$$
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 the ellipse equation, 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 the ellipse equation and the differential equation obtained by taking the derivative of the ellipse equation with respect to $x$, show that any ellipse in $E _ { u }$ satisfies
$$x y y ^ { \prime 2 } + \left( x ^ { 2 } - y ^ { 2 } - u ^ { 2 } \right) y ^ { \prime } - x y = 0 , \quad ( * * * )$$
where $y ^ { \prime } = \frac { \mathrm { d} y } { \mathrm {~d} x }$.
(3) As for the hyperbola equation, 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 $p = \frac { \mathrm { d} \beta } { \mathrm { d} \alpha }$.

Answer the following questions.
(1) Let $X$ be a real-valued random variable. Let $t$ be a real-valued variable. We define $\phi _ { X } ( t )$ for $X$ as
$$\phi _ { X } ( t ) = E _ { X } \left[ e ^ { t X } \right]$$
where $E _ { X } [ \cdot ]$ denotes the expectation taken with respect to $X$. Supposing that $\phi _ { X } ( t )$ is finite in a neighborhood of $t = 0$, give the mean and variance of $X$ using $\phi _ { X } ^ { \prime } ( 0 )$ and $\phi _ { X } ^ { \prime \prime } ( 0 )$. Here $\phi _ { X } ^ { \prime } ( t )$ and $\phi _ { X } ^ { \prime \prime } ( t )$ denote the first- and second-order derivatives of $\phi _ { X } ( t )$ with respect to $t$, respectively.
(2) For a sequence of mutually independent random variables: $X _ { 1 } , X _ { 2 } , \ldots , X _ { N }$, 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
$$p \left( X _ { j } = x \right) = \frac { 1 } { \sqrt { 2 \pi } \sigma } \exp \left( - \frac { ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } \right) .$$
Then calculate $\phi _ { X _ { j } } ( t )$. 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 $\phi _ { Z } ( t ) = \phi _ { W } ( t )$, the probability distribution of $Z$ is the same as that of $W$.
(3) Suppose that $N \in \{ 1,2 , \ldots , \infty \}$ 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
$$P ( N = n ) = ( 1 - \theta ) ^ { n - 1 } \theta$$
For $Y = X _ { 1 } + X _ { 2 } + \cdots + X _ { N }$, define $\phi _ { Y } ( t )$ by
$$\phi _ { Y } ( t ) = E _ { Y } \left[ e ^ { t Y } \right]$$
Then calculate $\phi _ { Y } ( t )$ and express it using $\phi _ { X _ { j } } ( t )$. Since $\phi _ { X _ { j } } ( t )$ does not depend on $j$, you can write it as $\phi _ { X } ( t )$.
(4) Calculate the mean and variance of $Y$ in Question (3).
(5) For given $\xi \left( > E _ { Y } [ Y ] \right)$, 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).