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Sean711822 · 2024年03月03日

看不懂这题在考什么？

  1. FRM I
  2. Quantitative

NO.PZ2020010304000016

问题如下：

Suppose that the annual profit of two firms, one an incumbent (Big Firm, X1) and the other a startup (Small Firm, X2), can be described with the following probability matrix:

What are the conditional expected profit and conditional standard deviation of the profit of Big Firm when Small Firm either has no profit or loses money (X2 ≤ 0)?

选项：

A.

3.01; 30.52

B.

3.01; 931

C.

1.03; 30.52

D.

1.03; 931.25

解释：

We need to compute the conditional distribution given X2 ≤ 0. The relevant rows of the probability matrix are

The conditional distribution can be constructed by summing across rows and then normalizing to sum to unity. The non-normalized sum and the normalized version are

Finally, the conditional expectation is E[X1|X2 ≤0] = Σx1Pr(X1 = x1|X2 ≤0) = USD 3.01M.

The conditional expectation squared is E[X1^2|X2 ≤0]=940.31, and so

the conditional variance is V[X1] = E[X1^2] - E[X1]^2 =940.31-3.01^2=931.25

and the conditional standard deviation is USD 30.52M.

normalize好像讲义里都没提到

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2 个答案

pzqa39 · 2024年03月14日

嗨，爱思考的PZer你好：


回复尚晨同学，没有写错，（5.87%+27.43%+13.5%+3.09%）*X=100%，解得X=2.0044097。再用这个数字分别去乘。5.87%*2.0044097=11.77%，27.43%*2.0044097=54.98%，13.5%*2.0044097=27.06%，3.09%*2.0044097=6.19%，算出来就是normalized那列数据。

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

pzqa39 · 2024年03月03日

嗨，爱思考的PZer你好：


non-normalized 就是概率直接相加，比如5.87%=1.97%+3.9%。normalize其实就是把non-normalized4个数等比例扩充到总概率等于100%就行了，假设一个扩大系数X，（5.87%+27.43%+13.5%+3.09%）*X=100%，求出来X，然后把这4个数分别乘以X就得到右边那列normalized数据了。

但是这道题虽然看起来很复杂，真正的考点是在得到解析的第二张图之后，主要只考了两个点：1、通过已知条件求条件概率，2、求方差的基本公式。

在解析第二张图的基础上：

E[X1|X2 ≤0]=-50*11.77%+0*54.98%+10*27.06%+100*6.19%=3.01

E[X1^2|X2 ≤0]是用-50，0，10，100分别取平方，然后再分别跟11.77%，54.98%，27.06%以及6.19%相乘，得到940.31

最后利用公式V[X1] = E[X1^2] - E[X1]^2 =940.31-3.01^2=931.25得到方差，再开根号得到最后的结果。

----------------------------------------------
就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

尚晨 · 2024年03月14日

第一段最后一句应该是5.87%除以X吧 是不是写错了 乘以的话怎么也算不出来normalized的那列数据啊

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