NO.PZ2016062402000003
问题如下:
The joint probability distribution of random variables X and Y is given by f{(x,y)}=k * x * y for x = 1,2,3, y = 1,2,3, and k is a positive constant. What is the probability that X+Y will exceed 5?
选项:
A.1/9
B.1/4
C.1/36
D.Cannot be determined
解释:
The function x*y is described in the following table. The sum of the entries is 36. The scaling factor k must be such that the total probability is 1. Therefore, we have k=1/36. The table shows one instance where X+Y > 5,which is x=3, y=3. The probability is p = 9/36 = 1/4.
中文解析:
这道题目考察的是联合概率分布。
符合题目条件的只有一种情况(X=3,Y=3),但要想求概率,就得首先求到联合概率分布公式中的k。
题目里说,它的联合概率分布是F(x,y)=k*x*y,x可以取3种,y可以取3种,所有总共有9种可能性。其中,每一种可能性的概率都为x*y*k。将9种可能性的概率相加,它的概率之和为36k。因为所有情况的概率之和为1,所以有36k=1,所以k=1/36。
最后P(X+Y>5)
= P(X=3, Y=3) = 9k =1/4
不懂为什么是1/36,函数不是k*x*y?x有3种可能性;y有3种可能性,不应该是1/9吗?