Science:Math Exam Resources/Courses/MATH103/April 2013/Question 06 (b)

MATH103 April 2013

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Question 06 (b)

Light Show on the Dance Floor A laser is mounted 2m above the floor (see sketch) and its beam is directed along a straight line on the dance floor. For visual effects, the angle $\theta$ changes periodically and randomly. The angle $\theta$ is chosen uniformly in the interval $(-{\frac {\pi }{4}},{\frac {\pi }{4}})$, that is, the probability density function (pdf) is $p(\theta )={\frac {2}{\pi }}$ for $-{\frac {\pi }{4}}<\theta <{\frac {\pi }{4}}$ and zero elsewhere. Find the probability density function (pdf) for $\displaystyle p(x)$ of the x-coordinate of the laser point on the floor.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What is the relationship between the cdf in terms of x and the pdf in terms of x.

Hint 2
The pdf in terms of x is just the derivative of the cdf in terms of x.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Taking the derivative of the cdf found in part a we have ${\begin{aligned}p(x)&=F'(x)=\left({\frac {2}{\pi }}\left(\arctan \left({\frac {x}{2}}\right)+{\frac {\pi }{4}}\right)\right)'\\&={\frac {2}{\pi }}\cdot \left({\frac {1}{1+\left({\tfrac {x}{2}}\right)^{2}}}\cdot {\frac {1}{2}}+0\right)\\&={\frac {1}{\pi }}\cdot {\frac {4}{4+x^{2}}}={\frac {4}{\pi (4+x^{2})}}\end{aligned}}$ completing the question.

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