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?
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everone Else!)
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!
Some good books to learn the art of probability
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
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Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R
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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.
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