In a class with 18 boys and 12 girls, boys have probability 1/3 of knowing the answer and girls have probability 1/2 of knowing the answer to a typical question the teacher asks. Assuming that whether or not the students know the answer are independent events, find the variance of the number of students who know the answer.

Answer: 7

Answer: 7

Let bi be the random variable that the ith boy knows the answer. Similarly define gi. This implies that

ReplyDeleteE(b1+...+b18+g1+...+g12)=12

E[(b1+...+b18+g1+...+g12)^2]=sum_{i} E[bi^2] + sum_{j} E[gj^2]+2sum_{i<j} E[bi*bj]+2sum_{i<j} E[gi*gj]+2*sum_{i,j} E[bi*gj] = 6+6+2/9*binomial(18,2)+2/4*binomial(12,2)+2*1/3*1/2*18*12=151

Hence

variance=151-12^2=7

This was much simpler than this. But the indicator function method rocks - I have grown to appreciated the power of this method.

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