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8527 · 2021年12月22日

b 题型

  1. FRM I
  2. Quantitative

NO.PZ2020010301000005

问题如下：

Based on the probabilities in the plot below, what are the values of the following?

a. Pr(AC)Pr(A^C)

b. Pr(D|A∪B∪C)

c. Pr(A|A)

d. Pr(B|A)

e. Pr(C|A)

f. Pr(D|A)

g. Pr((A∪D)C)Pr({(A∪D)}^C)

h. Pr((AC⋂DC))Pr((A^C\bigcap D^C))

i. Are any of the four events pairwise independent?

选项：

解释：

a. 1 - Pr(A) = 100% - 30% = 70%

b. This value is Pr(D∩(A∪B∪C))/Pr(A∪B∪C). The total probability in the three areas A, B, and C is 73%. The overlap of D with these three is 9% + 8% + 7% = 24%, and so the conditional probability is 24%/73%= 33%.

c. This is trivially 100%.

d. Pr(B∩A) = 9%. The conditional probability is 9%/30% = 30%.

e. There is no overlap and so Pr(C∩A) = 0.

f. Pr(D∩A) = 9%. The conditional probability is 30%.

g. This is the total probability not in A or D. It is 1 – Pr(A∪D) = 1 - (Pr(A) + Pr(D) - Pr(A∩D)) = 100% - (30% + 36% - 9%) = 43%.

h. This area is the intersection of the space not in A with the space not in D. This area is the same as the area that is not in A or D, Pr((A∪D)C)Pr({(A\cup D)}^C) and so 43%.

i. The four regions have probabilities A = 30%, B = 30%, C = 28% and D = 36%. The only region that satisfied the requirement that the joint probability is the product of the individual probabilities is A and B because Pr(A∩B) = 9% = Pr(A)Pr(B) = 30% * 30%.

请问像b小题这种类型在视频里哪里有讲到

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8527 · 2021年12月22日

根据这道题，麻烦老师再回答一下两个问题： 1. 请问 P(A n B) 的意思是 P（A|B), 还是A和B两合集中intersection的部分 ->P(AB). 如果答案是后者的话，为什么b小题的答案中‘’Pr(D|A∪B∪C)‘’ 等于 Pr(D∩(A∪B∪C))/Pr(A∪B∪C)。（n等于conditional ‘|’的概念） 麻烦老师解释一下什么是pairwise independent。 谢谢~！

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DD仔_品职助教 · 2021年12月22日

嗨，从没放弃的小努力你好：


同学你好，

b问是贝叶斯公式的变形，在基础班section1最后一个视频有讲。


1，P(AnB)A交B，代表他俩的交集。P(AuB)A并B，代表并集，所有面积，概率如下图绿色部分所示：

P(A|B)表示在已知A发生的概率下B发生的概率，具体计算公式有下面两种，这道题考的是第二个公式，第二个公式讲义里没给，补充记忆一下即可：

Pr(D|A∪B∪C)= Pr(D∩(A∪B∪C))/Pr(A∪B∪C)=D与ABC并集的交集/ABC的并集=24%/73%= 33%.


2，pairwise independent的英文解释是each event is independent of of every other possible combination of paired events.

翻译一下这个题中的ABCD这四个事件，我们任选两个事件，也就是一对，这对事件相对于其他时间，是不是独立的。

问的还是独立不独立，只不过针对的是两个事件。


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