(1) 已知随机变量 $X$ 的概率分布为
$$
P(X=1)=0.2, P(X=2)=0.3, P(X=3)=0.5
$$

试写出 $X$ 的分布函数 $F(x)$.
(2) 求 $X$ 的数学期望与方差.
(3) 已知随机变量 $\boldsymbol{Y}$ 的概率密度为
$$
f(y)= \begin{cases}\frac{y}{a^2} e^{-\frac{y^2}{2 a^2}} & y \geq 0 \\ 0 & y < 0\end{cases}
$$

求随机变量 $Z=\frac{1}{Y}$ 的数学期望 $E(Z)$.

已知某商品的需求量 $D$ 和供给量 $S$ 都是价格 $p$ 的函数:
$$
D=D(p)=\frac{a}{p^2}, S=S(p)=b p ，
$$

其中 $a>0$ 和 $b>0$ 是常数. 价格 $p$ 是时间 $t$ 的函数，且满足方程 $\frac{\mathrm{d} p}{\mathrm{~d} t}=k[D(p)-S(p)],(k$ 是常数 $)$ ，假设当 $t=0$ 时价格为 1 . 试求:
(1) 需求量等于供给量时的均衡价格 $P_e$;
(2) 价格函数 $p(t)$ ；
(3) 极限 $\lim _{t \rightarrow \infty} p(t)$.

Let $G$ be a finite group.
(1) Let $K$ be a field. Show that $G$ has a finite-dimensional fait hful $K$-linear representation.
(2) Show that $G$ has a faithful one-dimensional complex repr esentation if and only if $G$ is cyclic.
(3) Assume moreover that $G$ is commutative. Let $n \geq 1$ be an integer. Show that $G$ has a faithful $n$-dimensional complex re presentation if and only if $G$ can be generated by $n$ elements.
(4) Classify all finite groups having a faithful 2-dimensional re al representation.

Let $n \geq 1$ be an integer and write $\Phi_n(X)$ the $n$-th cyclotomi c polynomial, that is, the minimal polynomial of a primitive $n$ th root of unity in $\mathbb{C}$ over $\mathbb{Q}$. Write also $\varphi(n)=\operatorname{deg}\left(\Phi_n(X)\right)$
(1) Let $q$ be a power of a prime number such that $(q, n)=1$. Show that $\Phi_n$, viewed as an element in $\mathbb{F}_q[X]$, can be decom posed as a product of $\varphi(n) / d$ irreducible polynomials of deg ree $d$, with $d$ the order of $q$ in the multiplicative group $(\mathbb{Z} / n \mathbb{Z})^{\times}$.
(2) From now on, assume $n=2^{r+1}$ for some integer $r \geq 1$. L et $\zeta=\zeta_n$ be a primitive $n$-th root of unity and $K=\mathbb{Q}[\zeta]$. Let $p$ be a prime with $p \equiv-3(\bmod 8)$.
(a) For $x, y \in K=\mathbb{Q}[\zeta]$, define
$$
(x, y):=\sum_\tau \tau(x) \cdot \overline{\tau(y)}
$$
where $\tau$ runs through all the embeddings $K \hookrightarrow \mathbb{C}$ of $K$ into the field $\mathbb{C}$ of complex numbers. Write $K_{\mathbb{R}}=K \otimes_{\mathbb{Q}} \mathbb{R}$, and we use the same notation to denote the (à priori $\mathbb{C}$-valued) bi linear form on $K_{\mathbb{R}}$ obtained by extension of scalars. Show tha $\mathrm{t}(\cdot, \cdot)$ gives an inner product on $K_{\mathbb{R}}$ and for $0 \leq i, j < 2^r$,
$$
\left(\zeta^i, \zeta^j\right)= \begin{cases}2^r, & \text { if } i=j \\ 0, & \text { otherwise }\end{cases}
$$

In particular, we obtain an Euclidean space $K_{\mathbb{R}}$ and $\left(\zeta^i / \sqrt{2^r}\right)_{0 \leq i < 2^r}$ is an orthonormal basis.
(b) Decompose $p \mathcal{O}_K$ into a product of prime ideals.
(c) Let $\mathfrak{p} \subset \mathcal{O}_K$ be a prime ideal of $\mathcal{O}_K$ containing $p$. Show th at for every $\alpha \in \mathfrak{p},|\alpha|^2 \in 2^r p \mathbb{Z}$, and compute the length of $\mathrm{t}$ he shortest non-zero vector in the prime ideal $\mathfrak{p} \subset K_{\mathbb{R}}$.

Consider the integral
$$
\int_0^{\infty} f(x) \mathrm{d} x
$$
where $f$ is continuous, $f^{\prime}(0) \neq 0$, and $f(x)$ decays like $x^{-1-\alpha}$ with $\alpha>0$ in the limit $x \rightarrow \infty$.
(a) Suppose you apply the equispaced composite trapezoid $\mathrm{r}$ ule with $n$ subintervals to approximate
$$
\int_0^L f(x) \mathrm{d} x
$$

What is the asymptotic error formula for the error in the limit $n \rightarrow \infty$ with $L$ fixed?
(b) Suppose you consider the quadrature from (a) to be an ap proximation to the full integral from 0 to $\infty$. How should $L$ in crease with $n$ to optimize the asymptotic rate of total error $d$ ecay? What is the rate of error decrease with this choice of $L$ ?
(c) Make the following change of variable $x=\frac{L(1+y)}{1-y}$, $y=\frac{x-L}{x+L}$ in the original integral to obtain
$$
\int_{-1}^1 F_L(y) \mathrm{d} y
$$

Suppose you apply the equispaced composite trapezoid rule; what is the asymptotic error formula for fixed $L$ ?
(d) Depending on $\alpha$, which method - domain truncation of ch ange-of-variable - is preferable?

Consider the diffusion equation
$$
\frac{\partial v}{\partial t}=\mu \frac{\partial^2 v}{\partial x^2}, \quad v(x, 0)=\phi(x), \quad \int_a^b v(x, t) \mathrm{d} x=0
$$
with $x \in[a, b]$ and periodic boundary conditions. The solutio $\mathrm{n}$ is to be approximated using the central difference operator $L$ for the 1D Laplacian.
$$
L v_m=\frac{v_{m+1}-2 v_m+v_{m-1}}{h^2}
$$
and the following two finite different approximations, (i) Forw ard-Euler
$$
v_{n+1}=v_n+\mu k L v_n,(1)
$$
and (ii) Crank-Nicolson
$$
v_{n+1}=v_n+\mu k\left(L v_n+L v_{n+1}\right)
$$

Throughout, consider $[a, b]=[0,2 \pi]$ and the finite differenc e stencil to have periodic boundary conditions on the spatial lattice $[0, h, 2 h, \cdots,(N-1) h]$ where $h=\frac{2 \pi}{N}$ and $N$ is ev en.
(a) Determine the order of accuracy of the central difference operator $L v$ is approximating the second derivative $v_{x x}$.
(b) Using $v_m^n=\sum_{l=0}^{N-1} \hat{v}_l^n \exp \left(-i \frac{2 \pi l m}{N}\right)$ give the updates $\hat{v}_l^{n+1}$ in terms of $\hat{v}_l^n$ for each of the methods, including the ca se $l=0$.
(c) Give the solution for $v_m^n$ for each method when the initial condition is $\phi(m \Delta x)=(-1)^m$.
(d) What are the stability constraints on the time step $k$ for ea ch of the methods, if any, in equation (1) and (2)? Show there are either no constraints or express them in the form $k \leq F(h, \mu)$