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1234567 · 2025年06月22日

列一下具体计算公式

NO.PZ2020010304000015

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:

If an investor owned 20% of Big Firm and 80% of Small Firm, what are the expected profits of the investor and the standard deviation of the investor’s profits?

选项:

A.

4.18; 8.39

B.

4.18; 70.39

C.

2.18; 8.39

D.

2.18; 70.39

解释:

The expected profits are E[P] = 20% * E[X1] + 80% * E[X2] = 4.18.

The variance of the portfolio is (20%)^2*Var[X1] + (80%)^2*Var[X2] + 2(20%)(80%)Cov[X1, X2]

This value is 0.04 * 1534.36 + 0.64 * 2.41 + 2 * 0.16 * 23.37 = 70.39, and so the standard deviation is USD 8.39M.

这里COV(x1,x2)怎么求出来的?公式列举一下

1 个答案

李坏_品职助教 · 2025年06月22日

嗨,努力学习的PZer你好:


cov(x1,x2)=E(x1*x2)- E(x1)* E(x2)


这个E(X1*X2), 就是你把表格里每一个概率数字,乘以这个概率数字对应的X1的值 再乘以X2的值,最后求和。

也就是E(X1*X2) = 1.97% * (-1)*(-50)+ 3.9%*0 + 0.8%*2*(-50)+...


而E(X1):

X1 = -50 m 的概率:1.97%+3.9%+0.8%+0.1% = 6.77%

X1 = 0 的概率:3.93%+23.5%+12.6%+2.99% = 43.02%

X1 = 10 m 的概率:0.8%+12.7%+14.2%+6.68% = 34.38%

X1 = 100 m 的概率:0%+3.09%+6.58%+6.16% = 15.83%

E(X1) = -50* 6.77%+0*43.02%+10*34.38%+100*15.83% = 15.883


E(X2)与E(X1)同理。


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