Suppose that three-dimensional vectors $\left( \begin{array} { c } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$ satisfy the equation
$$\left( \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \\ z _ { n + 1 } \end{array} \right) = A \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right) \quad ( n = 0,1,2 , \ldots )$$
where $x _ { 0 } , y _ { 0 } , z _ { 0 }$ and $\alpha$ are real numbers, and
$$A = \left( \begin{array} { c c c } 1 - 2 \alpha & \alpha & \alpha \\ \alpha & 1 - \alpha & 0 \\ \alpha & 0 & 1 - \alpha \end{array} \right) , \quad 0 < \alpha < \frac { 1 } { 3 }$$
Answer the following questions.
(1) Express $x _ { n } + y _ { n } + z _ { n }$ using $x _ { 0 } , y _ { 0 }$ and $z _ { 0 }$.
(2) Obtain the eigenvalues $\lambda _ { 1 } , \lambda _ { 2 }$ and $\lambda _ { 3 }$, and their corresponding eigenvectors $\boldsymbol { v } _ { \mathbf { 1 } } , \boldsymbol { v } _ { \mathbf { 2 } }$ and $\boldsymbol { v } _ { \mathbf { 3 } }$ of the matrix $A$.
(3) Express the matrix $A$ using $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \boldsymbol { v } _ { 1 } , \boldsymbol { v } _ { 2 }$ and $\boldsymbol { v } _ { 3 }$.
(4) Express $\left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$ using $x _ { 0 } , y _ { 0 } , z _ { 0 }$ and $\alpha$.
(5) Obtain $\lim _ { n \rightarrow \infty } \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$. (6) Regard
$$f \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right) = \frac { \left( x _ { n } , y _ { n } , z _ { n } \right) \left( \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \\ z _ { n + 1 } \end{array} \right) } { \left( x _ { n } , y _ { n } , z _ { n } \right) \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right) }$$
as a function of $x _ { 0 } , y _ { 0 }$ and $z _ { 0 }$. Obtain the maximum and the minimum values of $f \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$, where we assume that $x _ { 0 } ^ { 2 } + y _ { 0 } ^ { 2 } + z _ { 0 } ^ { 2 } \neq 0$.

A real-valued function $u ( x , t )$ is defined in $0 \leq x \leq 1$ and $t \geq 0$. Here, $x$ and $t$ are independent. Suppose solving the following partial differential equation:
$$\frac { \partial u } { \partial t } = \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } ,$$
under the following conditions:
$$\begin{aligned} \text { Boundary condition : } & u ( 0 , t ) = u ( 1 , t ) = 0 , \\ \text { Initial condition : } & u ( x , 0 ) = x - x ^ { 2 } . \end{aligned}$$
Since the constant function $u ( x , t ) = 0$ is obviously a solution of the partial differential equation, consider the other solutions. Answer the following questions.
(1) Calculate the following expression, where $n$ and $m$ are positive integers.
$$\int _ { 0 } ^ { 1 } \sin ( n \pi x ) \sin ( m \pi x ) \mathrm { d } x$$
(2) Suppose $u ( x , t ) = \xi ( x ) \tau ( t )$, where $\xi ( x )$ is a function only of $x$ and $\tau ( t )$ is a function only of $t$. Express the ordinary differential equations for $\xi$ and $\tau$ using an arbitrary constant $C$. You may use that $f ( x )$ and $g ( t )$ are constant functions when $f ( x )$ and $g ( t )$ satisfy $f ( x ) = g ( t )$ for arbitrary $x$ and $t$.
(3) Solve the ordinary differential equations in question (2). Next, show that a solution of partial differential equation $(*)$ which satisfies the boundary condition is given by the following $u _ { n } ( x , t )$, and express $\alpha$ and $\beta$ using a positive integer $n$.
$$u _ { n } ( x , t ) = e ^ { \alpha t } \sin ( \beta x )$$
(4) The solution of partial differential equation $(*)$ which satisfies the boundary and initial conditions is represented by the linear combination of $u _ { n } ( x , t )$ as shown below. Obtain $c _ { n }$. You may use the result of question (1).
$$u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } u _ { n } ( x , t )$$

(1) If the probability density function $f ( t )$ of a continuous random variable $T$ is denoted by
$$f ( t ) = \begin{cases} \lambda e ^ { - \lambda t } & ( t \geq 0 ) \\ 0 & ( t < 0 ) \end{cases}$$
with a positive constant $\lambda$, then we say that $T$ follows an exponential distribution with parameter $\lambda$. Compute the average of this random variable. Also, derive the probability distribution function $F ( t ) = P ( T \leq t )$ of this exponential distribution, where $P ( X )$ is the probability of the event $X$.
(2) Show that the probability distribution given in question (1) is memoryless. Namely, show that
$$P ( T > s + t \mid T > s ) = P ( T > t )$$
holds for any $s > 0$ and $t > 0$, where $P ( X \mid Y )$ is the probability of the event $X$ conditioned on the event $Y$.
(3) Let us call the time interval between the time when one starts solving a problem and the time when one finishes it "time required for solution". Assume that the time required for solution of each of $n$ students follows the exponential distribution with the same parameter $\lambda _ { 0 }$. Let all the $n$ students start solving the problem at the same time. Find the probability distribution function and the average of the time required for solution of the student who finishes solving the problem earliest. Here, the time required for solution of each student is mutually independent.
(4) Assume that the times required for solution of a student A and a student B follow the exponential distributions with parameters $\lambda _ { A }$ and $\lambda _ { B }$, respectively. Let the two students start solving the problem at the same time. Find the probability that student A finishes solving the problem earlier than student B.
(5) Let a smart student Hideo and other ten students start solving a problem at the same time. Assume that the time required for solution of each of all the students follows an exponential distribution, where the average time required for solution of each of all the students except Hideo is ten times longer than that of Hideo. Find the probabilities that Hideo finishes solving the problem first and fourth, respectively.