IV - Probability Distribution-4

Question 1

区分：

discrete random variable

continuous random variable

Answer 1

前者：结果数量是可数的，即使有无限多个。对每一个结果有一个可以衡量的概率

一个月中下雨天数

后者：结果是不可数的，即使有上下界限。对任意一个结果，其概率为零，只能求得区间概率。

一个月降雨量（1-1000毫升）

似乎主要是一个确定单位的问题

Question 2

3 ways of discribing

probability distribution

表达方式，描述的是什么

Answer 2

1. probability function: for discrete, p(x)
2. probability density function/pdf: for continuous, f(x)
3. cumulative distribution function/cdf F(x) = P (X <=x)

Question 3

binomial distribution:

Bernoulli trial

equation

Answer 3

Question 4

区分Bernulli实验

和二项式分布

相同点1，不同点2

Answer 4

• 相同：都是只有两种outcome
• 不同：1.Bernulli只做一次实验，二项式做n次

2. 两者期望值、方差不同

Question 5

Answer 5

Question 6

Answer 6

Question 7

continuous uniform distribution

有一个怎样的特点

公式表示

Answer 7

一条水平线段

Question 8

multivariate distribution

需要知道几个元素？3

Answer 8

1. n个mean
2. n个variance
3. n(n-1)/2个correlation

Question 9

normal distribution

性质4条

Answer 9

Question 10

confidence interval

需要记住的4组值

Answer 10

Question 11

辨析

正态分布中，+-1.96标准差，得95%置信区间

根据Chebyshev不等式，+-1.96标准差，得75%置信区间

Answer 11

Chebyshev是普世适用，得最低值（至少75%）

标准分布图形收得更紧

Question 12

standard (unit) normal distribution

definition, and

Z value

Answer 12

标准正态分布，以Z表示

μ=0， s=1

Z value = (X-μ) / s

Question 13

standarizing a random variable:

equation

Answer 13

Question 14

shortfall risk

definition

Answer 14

the risk a portfolio value will fall below the

客户要求的最低回报

Question 15

Roy’s

Safety First Ratio

equation

及推论

Answer 15

SFRatio 越大者越符合要求

Question 16

比较

SFRatio

&

Sharpe Ratio

Answer 16

当客户要求最低回报率就是无风险回报率时，

SF ratio = Sharpe ratio

Question 17

lognormal distribution

1。定义

2。图形左右偏

3。极限值

4。应用

Answer 17

1. if LnX is normal, then X is lognormal
2. right / positively skewed
3. 0 - 正无穷
4. describe asset price

Question 18

normal distriution

lognormal distribution

分别用来描述

Answer 18

1. returns
2. share and asset price

Question 19

if continuously compounded returns are normally distributed

Answer 19

then asset prices are lognormally distributed

Question 20

lognormal distridution:

已知连续复利回报HPR，求名义利率r

Answer 20

e^r - 1 = HPR

e^r = HPR + 1

r = ln (HPR+1)

Question 21

Monte Carlo Simulation

描述1 点 区别 2点

Historical Simulation

Answer 21

1. M对风险因素进行虚拟建模，H是对过去实际发生的风险因素进行统计
2. M的缺点：一是复杂，二是需根据输入的分布参数来进行模拟
3. H的缺点：未来的风险因素不一定等于过去的，因此H不能解决“what if”的问题。选取的时间段若未包含罕见的极端情况，则无法反映在统计结果中

Question 22

For a binomial random variable with five trials,

and a probability of success on each trial of 0.5,

the distribution will be:

Answer 22

Symmetric.

Other than P=0.5, the distribution would be asymmetric

Question 23

The total number of parameters that fully characterizes a multivariate normal distribution for the returns on two stocks is :

Answer 23

5.

two means, two variables, and one correlation

Question 24

Monte Carlo simulation

实质是

Answer 24

providing a distribution of possible solutions to complex functions