NO.PZ2024120401000007
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:
Using the probability matrix for Big Firm and Small Firm, what is the correlation between the profits of these two firms?
选项：
A.0.384
B.0.350
C.23.37
D.43.7
解释：
The marginal distributions are computed by summing across rows for Big Firm and down columns for Small Firm. They are:
The variance can be computed using E[Xj2]- (E[Xj])2 .
For Big Firm, these values are E[X1] = 6.77% * (-50) + 43.02% * (0) + 34.38% * (10) + 15.83% * (100) = $15.88M, E[X12] = 1786.63 and V[X1] = 1534.36.
For Small Firm, these values are E[X2] = 6.70% * (-1) + 43.19% * (0) + 34.18% * (2) + 15.93% * (4) = $1.25M, E[X22] = 3.98 and V[X2] = 2.41.
The expected value of the cross product is E[X1X2] = ΣΣx1x2Pr(X1 = x1, X2 = x2) = 43.22.
The covariance is E[X1X2]- E [X1]×E[X2] = 43.22- 15.88 * 1.25 = 23.37 and the correlation is 23.37 / sqrt(2.41 * 1534.36) = 0.384.
不明白"correlation" 問的是什麼，然後引伸至解題的步驟
另外，不明白的步驟，尤其是：
The expected value of the cross product is E[X1X2] = ΣΣx1x2Pr(X1 = x1, X2 = x2) = 43.22.
The covariance is E[X1X2]- E [X1]×E[X2] = 43.22- 15.88 * 1.25 = 23.37 and the correlation is 23.37 / sqrt(2.41 * 1534.36) = 0.384.
