QM5_Didstribution

Basic Concepts

Probability distribution

Discrete distribution (离散分布)

• The distribution of the discrete random variable.
• Discrete random variable
  • takes on a finite and countable number of possible values.

Continuous distribution (连续分布)

• The distribution of the continuous random variable.
• Continuous random variable:
  • takes on an infinite and uncountable number of possible values.

Probability function

Probability function (概率函数)

• For discrete random variable taking on a specific value
• p(x) = P(X=x)
  • X: 随机变量; x: a specific value

Probability density function (概率密度函数)

• For continuous random variables within a range
• P(x1<X<x2)
• The probability of taking on an specific value is always zero, P(X=x)=0
• 因为连续随机变量有无数个数, 即分母为无穷大, 所以取到一个具体的值的概率是0. 这并不代表不可能取到个这值,只是相对于取值范围, 取到该值的可能性太小. 所以对连续随机变量, 研究具体的值没有什么价值, 研究的是range.

Cumulative probability function (累积概率函数)

• F(x) = P(X<=x)

插入图片, 用面积代表概率.

Dsicrete uniform distribution (离散均匀分布)

• Definition
  • has a finite number of possible outcomes, all of which are equally likely.
• Example: X = {1,2,3,4,5}
  • P(1) = P(2) = P(3) = P(4) = P(5) = 20%
  • P(3) = P(1) + P(2) + P(3) = 60%
  • P(2<=X<=4) = P(2) + P(3) + P(4) = 60%

Discrete Distribution

Binomial distribution ** (二项式分布)

Bernoulli random variable (trial,伯努利实验)

• Random variables with only two outcomes, one represents success(denoted as 1); the other represents failure(denoted as 0). P(X=1) = p, P(X=0) = 1-P.
• Binomial random variable
  • The number of successes in a Bernoulli trials. (做n次Bernoulli trials就得到二项式分布)
  • The probability of x successes in n trails.

插入老师板书.插入公式.

• Expected value and variance

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• 均值 => 期望值 => 算加权平均
• 计算器算排列组合: 10个中挑出6个, 10 => 2nd => + ->6 -> =.

Continuous Distribution

Continuous uniform distribution (连续均匀分布)

• Definition
  • probability of continuous uniform random variable which distribute evenly over an interval.
• Properties
  • P(X=x)=0
  • P(x1<=X<=x2) = (x2-x1)/(b-a)

Normal distribution *** (正态分布)

• Properties

  • completely described by mean and variable.
    • 只由两个参数决定 , 均值和方差.

  • 插入公式

  • skewness = 0, kurtosis = 3

  • Linear combination of normally distributed random variables is also normally distributed.
    • 比如x1~n是线性的, x2~n也是线性的, 则3x1+2x2~n也是线性的.
  • Probability descrase further from the mean, but the tails go on forever.

• 考点

  • 性质
  • 置信区间
  • 标准化

• Concepts

  • Confidence interval 置信区间
    • 落在区间内的概率, 就是切比雪夫.
  • Confidence level 置信水平
    • 置信水平 = 置信度
  • Confidence degree 置信度
• Properties

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• k : 依赖因子(关键值), Reliability factor /Critical value.

• Standard normal distribution 标准正态分布

  • also named z-distribution
  • X~N (0,1), 正态分布均值是0, 方差是1.
  • Standardization
  • >>>插入公式
  • z值含义
    • 正态分布与标准正态分布位置相对应.
    • 离标准正态分布均值的距离,即z个标准差.
    • 算出z值后,查表(z-table), 查出累计概率.
• 例题

Shortfall risk **(缺口风险)

• Definition
  • the risk that portfolio value or return will fall below the imnimum acceptable level(RL)
• Properties
  • The lower, the better

Safety-first ratio **(第一安全比率)

• Definition
  • the distance from the mean return to the shortfall elvel in units of standard deviation.
• Calculation
• >>>插入公式
• Properties
• The higher, the better
• Minimizing shortfall risk - Maximizing safety-first ratio

Lognormal distribution **

• Properties

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Student;s t-distribution ***

• Properties
  • Defined by single parameter: degree of freedom(df), 由唯一一个参数决定.
    • df = n-1, where: n is the sample size.
  • Symmetrical, skewness = 0,
  • Fatter tails than a normal distribution (低峰肥尾)
  • As df increase t-distribution is approaching to standard normal distribution.
  • Given a degree of confidence, t-distribution has a wider confidence interval than z-distribution.
• Shape

插图

Simulation (模拟)

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