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magickame · 2021年07月07日

最好能把全过程算一下而不是简单列公式

NO.PZ2020010304000014

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 covariance and correlations between the profits of these two firms?

选项:

解释:

First, the two means and variances can be computed from the marginal distributions.

Then the covariance can be computed using the alternative form, which is E[X1X2]E[X1]E[X2]E[X_1X_2] - E[X_1]E[X_2].

The means can be computed using E[Xj]=ΣxjPr(Xj=xj)E[X_j] = Σx_j Pr(X_j = x_j) for j = 1,2.

The variance can be computed using E[Xj2](E[Xj])2E[X_j^2]-(E[X_j])^2, which requires computing E[Xj2]E[X_j^2] using Σxj2]Pr(Xj=xj)Σx_j^2]Pr(X_j = x_j).

For Big Firm, these values are E[X1]=USD15.88ME[X_1] = USD 15.88M, E[X12]=1786.63E[X_1^2] = 1786.63 and V[X1]=1534.36V[X_1] = 1534.36.

For Small Firm, these values are E[X2]=USD1.25ME[X_2] = USD 1.25M, E[X22]=3.98E[X_2^2] = 3.98 and V[X2]=2.41V[X_2] = 2.41.

The expected value of the cross product is E[X1X2]=ΣΣx1x2Pr(X1=x1,X2=x2)=43.22E[X_1X_2] = ΣΣx_1x_2Pr(X_1 =x_1, X_2 = x_2) = 43.22. The covariance is then 43.22 - 1.25 * 15.88 = 23.37 and the correlation is 23.37/(22.411534.36)=0.38423.37/(\sqrt{22.41 * 1534.36}) = 0.384

公式太复杂,全步骤计算详解出来理解更方便

2 个答案

品职答疑小助手雍 · 2022年01月17日

同学你好,题目让求的是cov和ρ,所以都要算。

品职答疑小助手雍 · 2021年07月07日

嗨,爱思考的PZer你好:


这道题看起来写了这么长正是因为解析写的很详尽,其实总共就用了三个基础公式:

一个是求期望的公式,另一个是协方差的公式cov=E[X1*X2] - E[X1]E[X2],方差的公式可以套用协方差的公式,X1的方差就是把cov公式里的X2都换成X1就行了。还有一个是通过协方差求correlation的公式。


计算过程答案里都写了,我再翻译一遍:

首先求两个数据单独的期望和方差:E(X1)=15.88M, E(X1的平方)=1786.63,X1的方差就是1786.63-15.88*15.88=1534.36

E(X2)=1.25M, E(X2的平方)=3.98,X2的方差就是3.98-1.25*1.25=2.41

E(X1*X2)=每个格子里的概率*X1*X2,再把所有格子的结果加起来,等于43.22。

所以cov=43.22-1.25*15.88=23.37。

correlation=cov/(X1的标准差*X2的标准差)也就是解析的最后一步,等于0.384。

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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!

Clare · 2022年01月17日

已经有cov了还需要算ρ吗

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NO.PZ2020010304000014问题如下 Suppose ththe annuprofit of two firms, one incumbent (Big Firm, X1) anthe other a startup (Small Firm, X2), cscribewith the following probability matrix:whare the covarianancorrelations between the profits of these two firms? First, the two means anvariances ccomputefrom the marginstributions.Then the covarianccomputeusing the alternative form, whiis E[X1X2]−E[X1]E[X2]E[X_1X_2] - E[X_1]E[X_2]E[X1​X2​]−E[X1​]E[X2​].The means ccomputeusing E[Xj]=ΣxjPr(Xj=xj)E[X_j] = Σx_j Pr(X_j = x_j)E[Xj​]=Σxj​Pr(Xj​=xj​) for j = 1,2.The varianccomputeusing E[Xj2]−(E[Xj])2E[X_j^2]-(E[X_j])^2E[Xj2​]−(E[Xj​])2, whirequires computing E[Xj2]E[X_j^2]E[Xj2​] using Σxj2]Pr(Xj=xj)Σx_j^2]Pr(X_j = x_j)Σxj2​]Pr(Xj​=xj​).For Big Firm, these values are E[X1]=US5.88ME[X_1] = US15.88ME[X1​]=US5.88M, E[X12]=1786.63E[X_1^2] = 1786.63E[X12​]=1786.63 anV[X1]=1534.36V[X_1] = 1534.36V[X1​]=1534.36.For Small Firm, these values are E[X2]=US.25ME[X_2] = US1.25ME[X2​]=US.25M, E[X22]=3.98E[X_2^2] = 3.98E[X22​]=3.98 anV[X2]=22.41V[X_2] = 22.41V[X2​]=22.41.The expectevalue of the cross prois E[X1X2]=ΣΣx1x2Pr(X1=x1,X2=x2)=43.22E[X_1X_2] = ΣΣx_1x_2Pr(X_1 =x_1, X_2 = x_2) = 43.22E[X1​X2​]=ΣΣx1​x2​Pr(X1​=x1​,X2​=x2​)=43.22. The covarianis then 43.22 - 1.25 * 15.88 = 23.37 anthe correlation is 23.37/(22.41∗1534.36)=0.38423.37/(\sqrt{22.41 * 1534.36}) = 0.38423.37/(22.41∗1534.36​)=0.384 图中的1m2m50m是啥意思啊😰题干看不懂

2024-03-13 13:45 1 · 回答

NO.PZ2020010304000014 问题如下 Suppose ththe annuprofit of two firms, one incumbent (Big Firm, X1) anthe other a startup (Small Firm, X2), cscribewith the following probability matrix:whare the covarianancorrelations between the profits of these two firms? First, the two means anvariances ccomputefrom the marginstributions.Then the covarianccomputeusing the alternative form, whiis E[X1X2]−E[X1]E[X2]E[X_1X_2] - E[X_1]E[X_2]E[X1​X2​]−E[X1​]E[X2​].The means ccomputeusing E[Xj]=ΣxjPr(Xj=xj)E[X_j] = Σx_j Pr(X_j = x_j)E[Xj​]=Σxj​Pr(Xj​=xj​) for j = 1,2.The varianccomputeusing E[Xj2]−(E[Xj])2E[X_j^2]-(E[X_j])^2E[Xj2​]−(E[Xj​])2, whirequires computing E[Xj2]E[X_j^2]E[Xj2​] using Σxj2]Pr(Xj=xj)Σx_j^2]Pr(X_j = x_j)Σxj2​]Pr(Xj​=xj​).For Big Firm, these values are E[X1]=US5.88ME[X_1] = US15.88ME[X1​]=US5.88M, E[X12]=1786.63E[X_1^2] = 1786.63E[X12​]=1786.63 anV[X1]=1534.36V[X_1] = 1534.36V[X1​]=1534.36.For Small Firm, these values are E[X2]=US.25ME[X_2] = US1.25ME[X2​]=US.25M, E[X22]=3.98E[X_2^2] = 3.98E[X22​]=3.98 anV[X2]=2.41V[X_2] = 2.41V[X2​]=2.41.The expectevalue of the cross prois E[X1X2]=ΣΣx1x2Pr(X1=x1,X2=x2)=43.22E[X_1X_2] = ΣΣx_1x_2Pr(X_1 =x_1, X_2 = x_2) = 43.22E[X1​X2​]=ΣΣx1​x2​Pr(X1​=x1​,X2​=x2​)=43.22. The covarianis then 43.22 - 1.25 * 15.88 = 23.37 anthe correlation is 23.37/(22.41∗1534.36)=0.38423.37/(\sqrt{22.41 * 1534.36}) = 0.38423.37/(22.41∗1534.36​)=0.384 实际考试会让计算E(x1y1)吗?计算器如何快速按出来呢?还是对于3*3的格子要按9遍?

2022-09-07 08:32 1 · 回答

看了解析,没明白,希望用中文再一步步把答案分析出来

2020-02-21 22:35 3 · 回答

请问能拆解一下E[X1​X2​]=ΣΣx1x2Pr(X1​=x1​,X2=x2​)=43.22吗?

2020-02-14 14:11 1 · 回答