Q: You ask two students A & B to do a statistical task. The task is to roll two dies, sum the numbers and to repeat it a 100 times. You get back the set of numbers from both the students. However you know that one of them is a lazy student and has rolled just one die and doubled its value and reported it. How do you identify which one of the students is the lazy student?

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A: Let \(X_{1}\) and \(X_{2}\) represent the random variables that shows up when a die is rolled. This can be any of 1 to 6. The sum of the two random variables would have a variance given as

$$

Var(X_{1} + X_{2}) = Var(X_{1}) + Var(X_{2})

$$

If we let \(Var(X_{1}) = Var(X_{2}) = \alpha\) then \(Var(X_{1}) + Var(X_{2}) = 2\alpha\). The lazy student doubles the die score. The variance for a multiplier to the random variable works out as

$$

Var(\beta X) = \beta^{2}Var{X}

$$

So the lazy student's variance would work out to be \(2^2\alpha = 4\alpha\). Note that the variance of the lazy student is double that of the regular student. By simply computing the variance of the scores, one can make a reasonable guess at identifying the lazy student!

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Almost always (in terms of the two returned sets) you can identify the lazy student with absolute certainty: his report would contain only even numbers.

ReplyDeleteAlmost always (in terms of the two returned sets) you can identify the lazy student with absolute certainty: his report would contain only even numbers.

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