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🌊 梅根 · 2020年02月14日

问一道题：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 $E[X_1X_2] - E[X_1]E[X_2]$ .

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

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

For Big Firm, these values are $E[X_1] = USD 15.88M$ , $E[X_1^2] = 1786.63$ and $V[X_1] = 1534.36$ .

For Small Firm, these values are $E[X_2] = USD 1.25M$ , $E[X_2^2] = 3.98$ and $V[X_2] = 2.41$ .

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

请问能拆解一下E[X1X2]=ΣΣx1x2Pr(X1=x1,X2=x2)=43.22吗？

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品职答疑小助手雍 · 2020年02月14日

同学你好，E（x1*x2）等于表格里每个格子的概率*概率对应的x1*x2，拿第一格举例就是1.97%*1M*（-50M），最后加总等于43.22

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任会会 · 2020年03月13日

为什么我算出来不是43.22，而是28.47呢

品职答疑小助手雍 · 2020年03月13日

把带0的项去除就8个格子，我放在excel算完了是43.22啊

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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[X1X2]−E[X1]E[X2].The means ccomputeusing E[Xj]=ΣxjPr(Xj=xj)E[X_j] = Σx_j Pr(X_j = x_j)E[Xj]=ΣxjPr(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[X1X2]=ΣΣx1x2Pr(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[X1X2]−E[X1]E[X2].The means ccomputeusing E[Xj]=ΣxjPr(Xj=xj)E[X_j] = Σx_j Pr(X_j = x_j)E[Xj]=ΣxjPr(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[X1X2]=ΣΣx1x2Pr(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

NO.PZ2020010304000014 公式太复杂，全步骤计算详解出来理解更方便

2021-07-07 10:41 2 · 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