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February 13th, 2013, 04:32 AM   #1
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Stochastic variables

I'm stuck on a set of problems, and was hoping for some help. Here's the problem:

We have a multiple choice test with 4 problems, each with 4 possible answers. Let X be the number of correct answers a random candidate gets, Y is the number of questions he actually knows the answer to, and Z is the number of question he guesses correctly. It follows that X = Y + Z.

Assume that if one guesses on one of the problems, it is completely random which one of the 4 alternatives is correct, and if one does know the answer, he will give the right answer.

a) Let's look at a person who knows the answer to one of the questions. Explain why Z in this case is Bin(3, 0.25). Calculate and E(Z). Also find E(X) for this person.


What I've got so far, is the first part. I've reasoned that since the person knows the answer to one of the questions, he needs to guess on the other 3. The probability of each of them being a correct guess is 0.25. Hence Z is binomially (?) distributed with n=3 and p=0.25.

But I'm stuck on the other parts. Can anyone help out? Thanks in advance!
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