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CC · 2024年02月18日

第一问3.01怎么算的

  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.

 E[X1|X2 ≤0] = Σx1Pr(X1 = x1|X2 ≤0) = USD 3.01M

根据公式算出来的是1.5050，是哪里有问题吗？



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2 个答案

品职答疑小助手雍 · 2024年02月25日

解析里第二个表格 non-normalized那一列就是没有标准化过的非条件概率，这一列概率加和等于49.89%。

标准化就是把这个49.89%当做全集100%，算那一列的四个概率占全集的多少。所以就是分别用那四个非标准化的概率除以49.89%得到右边那一列标准化的概率。

品职答疑小助手雍 · 2024年02月19日

同学你好，注意提问的开头有一个conditional，也就是说是在大公司的利润已经在0或以下的情况下，小公司收益的期望。所以应该把你写的括号里的概率当做一个全集（100%）来进行标准化后计算。

CC · 2024年02月20日

我的思路也是条件概率，只是没标准化，这还是标准化的问题。懂了，谢谢

王佳 · 2024年02月25日

想问标准化是怎么算出来的？

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