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.
本题考点为条件期望和条件标准差，涉及期望、标准差计算公式以及条件概率性质。计算步骤如下：
1、条件期望为 E[X1|X2 ≤0]= Σx1Pr(X1 = x1|X2 ≤0)= Σx1*[Pr(X1 = x1,X2 ≤0)/Pr(X2 ≤0)]，依次带入X1取4个概率的数值计算。
2、条件标准差为E[X1^2|X2 ≤0]- E[X1|X2 ≤0]^2，其中
E[X1^2|X2 ≤0]=Σx1^2*Pr(X1 = x1|X2 ≤0)=Σx1^2*[Pr(X1 = x1,X2 ≤0)/Pr(X2 ≤0)]；
E[X1|X2 ≤0]^2为计算1的结果。
想知道以上思路和计算过程是否正确；整个计算比较繁琐，容易出错，是否有更高效的计算方法？
