Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 5 (a)
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Question B 5 (a) 

The percentage of people with the disease, March Madness, is recorded every week. Note that it is possible to recover from March Madness one week and catch it again the following week. Records indicate that the disease can be modelled by a random walk and that if 50% of the population is infected with March Madness one week, then 60% of the population will be infected the next week. Records also indicate that if 100% of the population is infected one week, then 90% of the population will be infected the next week. It is known that 10% of the population has March Madness this week.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

To find the matrix, it is enough to recall that matrix multiplication is a linear transformation and that the columns of a matrix are the images of the standard basis vectors under that matrix. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Represent the proportion of people who are infected as column vectors like so:
And let be the desired transition matrix. The data given in this question becomes,
The first column of is and the second column of is . So, it is enough to find . We have that
So as is a linear transformation,
Hence,
