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维京飞🐤 · 2022年07月12日

您好，non_ normalized啥意思？还有

  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.

​还有整个思路能再详细一点吗？看不懂在算什么？

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品职答疑小助手雍 · 2022年07月12日

同学你好，non-normalized 就是概率直接相加，normalized 才是条件概率，就是使总和为100%

这种题不要觉得看到一个表格数字很多（其实表格可以看成一个韦恩图），解析算的多就很难；

它主要只考了两个点：1、通过已知条件求条件概率，2、求方差的基本公式。

条件概率和normalized条件概率就是按照解析里的第二张图求的，用求出来的概率求期望的平方和平方的期望这应该没啥问题，然后把期望的平方和平方的期望带入E[X^2] - E[X]^2这个公式里求方差，再开方求标准差就行了。

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