一阶微分方程

1. 如果微分方程是$\frac{\mathrm{d}x}{\mathrm{d}y} = Q(x)$，那么解就是$y=\int Q(x) \mathrm{d}x$。
2. 分离变量法。如果微分方程为$\frac{\mathrm{d}x}{\mathrm{d}y} = f(x)g(y)$，则可以将其改写为$\frac{1}{f(x)}\mathrm{d}x = g(y) \mathrm{d}y$，然后对每条边积分，即可得到解。
3. 积分因子. 如果微分方程为 $\frac{\mathrm{d}x}{\mathrm{d}y}+P(x)y=Q(x)y$, 那他就可以转化为 $$e^{\int P(x) \ \mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} + e^{\int P(x) \ \mathrm{d}x} \ P(x)y = e^{\int P(x) \ \mathrm{d}x} \ Q(x) $$ 通过每项都乘 $$e^{\int P(x) \ \mathrm{d}x}$$, 因此
   $$e^{\int P(x) \ \mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} + e^{\int P(x) \ \mathrm{d}x} \ P(x)y = e^{\int P(x) \ \mathrm{d}x} \ Q(x) \ $$
   然后使用积分的乘积法则, 则,
   $$ \frac{\mathrm{d}}{\mathrm{d}x} (e^{\int P(x) \ \mathrm{d}x} y) = e^{\int P(x) \ \mathrm{d}x} Q(x) $$

First Order Differential Equation

1. Simple one. If the differential equation is like $\frac{\mathrm{d}x}{\mathrm{d}y} = Q(x)$, then solution would be $y=\int Q(x) \mathrm{d}x$.
2. Separation of variables. If the differential equation is like $\frac{\mathrm{d}x}{\mathrm{d}y} = f(x)g(y)$, then it could be re-write in $\frac{1}{f(x)}\mathrm{d}x = g(y) \mathrm{d}y$, then integrating each side and then the solution can be goten.
3. Integrating factor. If the differential equation is like $\frac{\mathrm{d}x}{\mathrm{d}y}+P(x)y=Q(x)y$, then it could become $$e^{\int P(x) \ \mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} + e^{\int P(x) \ \mathrm{d}x} \ P(x)y = e^{\int P(x) \ \mathrm{d}x} \ Q(x) $$ by multiplying $$e^{\int P(x) \ \mathrm{d}x}$$, so that
   $$e^{\int P(x) \ \mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} + e^{\int P(x) \ \mathrm{d}x} \ P(x)y = e^{\int P(x) \ \mathrm{d}x} \ Q(x) \ $$
   using product rule,
   $$ \frac{\mathrm{d}}{\mathrm{d}x} (e^{\int P(x) \ \mathrm{d}x} y) = e^{\int P(x) \ \mathrm{d}x} Q(x) $$

Basic Concept

Unlike discrets probability distribution, continuous probability distribution is continuous. The variables of the former can only be integers, while the latter is the entire field of real numbers.

Similar to discrete notation, the probability is usually written as $P(X \leq x)$. However, it should be noted that for a continuous probability distribution, $P(X=x)=0$, that is, for a continuous probability distribution, the probability of obtaining a particular value is 0. Notes: the reference to “probability 0” here DOES NOT mean that it is impossible (it can be analogous to “taking a random point on the field of real numbers, the probability that the point is rational is 0”).

Basic Concept

与discrets probability distribution不同的是, continuous probability distribution是连续的. 前者的变量只能是整数, 而后者是整个实数域.

与离散的写法类似, 其概率通常写做 $P(X \leq x)$. 但需要注意的是, 对于一个continuous probability distribution来说, $P(X=x)=0$, 意味着对于一个连续的概率分布而言, 得到一个特定的值的概率为0. 需要注意的是, 在这里所指的“概率为0”并不意味着一定不可能出现(可以和“在实数域上随便取一点, 该点为有理数的概率为0”类比).

Basic Concept

$x$: random variable, where $x \in \mathbb{R} \quad or \quad x \in \mathbb{Z}$

$P(x=k)=a$: For the event $x=k$, his probability is $a$. Note that the probability of any event is greater than or equal to 0 and less than or equal to one.

for example, x: outcome of coin tossing. tail:0, head:1.
$$P(x=0)=\frac{1}{2}, \quad P(x=1)=\frac{1}{2}$$

Notes: for a discrete probability distribution, the sum of the probabilities of all events must be 1, i.e:
$$\sum_{i=-\infty}^{\infty} P(x=i) = 1$$

This is often used as a way of checking whether a probability distribution is valid.

Basic Concept

$A$, $B$: event:

• $P(A)$: probability that event A happens;
• $P(A \cap B)$: probability that both event A and B happen;
• $P(A \cup B)$: probability that event A or B happen (or both).

Inclusion–exclusion principle: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Mutually exclusive events: $A \cap B = \varnothing$, i.e. $P(A \cap B) = 0$.

A, B independent: A and B have no impact on each other, i.e. $P(A \cap B) = P(A) \cdot P(B)$.

$P(A’)=1-P(A)$: comlimentary

Basic Concept

$x$: random variable. 其中 $x \in \mathbb{R} \quad or \quad x \in \mathbb{Z}$

$P(x=k)=a$: 对于$x=k$这个事件, 他的概率为$a$. 需要注意, 任何事件的概率均大于等于0且小于等于一.

举个例子, x: outcome of coin tossing. tail:0, head:1.
$$P(x=0)=\frac{1}{2}, \quad P(x=1)=\frac{1}{2}$$

需要注意的是, 对于一个离散的概率分布, 所有事件的概率之和一定为1, 即:
$$\sum_{i=-\infty}^{\infty} P(x=i) = 1$$
这个通常会作为检验一个概率分布是否有效的方式.

Basic Concept

$A$, $B$: event(事件):

• $P(A)$: 事件A发生的概率;
• $P(A \cap B)$: 事件A和B同时发生的概率;
• $P(A \cup B)$: 事件A或B任意一个发生的概率.

容斥原理: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

互斥事件(mutuallu exclusive events): $A \cap B = \varnothing$, 即$P(A \cap B) = 0$.

独立事件(independent events): 事件A和B独立发生, 均不会对对方产生任何影响. $P(A \cap B) = P(A) \cdot P(B)$.

互补事件(complementary events): $P(A’)=1-P(A)$

Basic Concepts

Definition: $i=\sqrt{-1}$

Complex number: $a+bi$, where $a$ is real part，$bi$ is imaginary part. $a={\rm Re}(\mathbb{Z})$, $b={\rm Im}(\mathbb{Z})$

Notes: $i^2=-1$, $i^3=-i$, $i^4=1$

It can also be written in vector form, such as $a \choose b$ (Note: this writing is only useful when calculating addition and subtraction, other operations are more troublesome).

Basic Concepts

定义: $i=\sqrt{-1}$

复数(Complex number): $a+bi$, 其中$a$为实数部分，$bi$为虚数部分. $a={\rm Re}(\mathbb{Z})$, $b={\rm Im}(\mathbb{Z})$

注意: $i^2=-1$, $i^3=-i$, $i^4=1$

也可以用向量的形式书写, 如 $a \choose b$ (注: 这种写法只在计算加减法的时候好用，其他运算比较麻烦)