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

MATH103 April 2013

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Question 06 (c)

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. The mean x-coordinate $\displaystyle {\overline {x}}$ of the laser point is given by $\displaystyle {\overline {x}}=\int _{a}^{b}xp(x)\,dx$ where a corresponds to the x value for $\displaystyle \theta ={\frac {\pi }{4}}$ and b corresponds to the x value for $\displaystyle \theta ={\frac {-\pi }{4}}$. Evaluate this integral (or state that it diverges). NOTE: There was an error in the question as written on the exam and has been fixed on the wiki.

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
There is a clever solution by symmetry that you should exploit if you see it.

Hint 2
(Alternate solution) Otherwise, try a substitution.

Hint 3
A good substitution is by setting $\displaystyle u=4+x^{2}$.

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 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Notice that $\displaystyle xp(x)={\frac {2x}{4+x^{2}}}$ is an odd function. The interval in question comes from noting that $\displaystyle x=2\tan(\theta )$ and so $\displaystyle a=2\tan(-\pi /4)=-2$ and $\displaystyle b=2\tan(\pi /4)=2$ and so $\displaystyle [a,b]=[-2,2]$ is symmetric about the origin. Therefore, $\displaystyle {\overline {x}}=\int _{a}^{b}xp(x)\,dx=0$ by symmetry of an odd function.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. As in the previous solution, the interval in question comes from noting that $\displaystyle x=2\tan(\theta )$ and so $\displaystyle a=2\tan(-\pi /4)=-2$ and $\displaystyle b=2\tan(\pi /4)=2$. To solve the integral $\displaystyle {\overline {x}}=\int _{a}^{b}xp(x)\,dx=\int _{-2}^{2}xp(x)\,dx=\int _{-2}^{2}{\frac {2x}{4+x^{2}}}\,dx$ we use a substitution given by $\displaystyle u=4+x^{2}$ so that $\displaystyle du=2xdx$, $\displaystyle u(-2)=4+(-2)^{2}=8$, $\displaystyle u=4+(2)^{2}=8$ and so ${\begin{aligned}{\overline {x}}&=\int _{a}^{b}xp(x)\,dx=\int _{-2}^{2}xp(x)\,dx\\&=\int _{-2}^{2}{\frac {2x}{4+x^{2}}}\,dx\\&=\int _{8}^{8}{\frac {du}{u}}\\&=\ln |u|{\big |}_{8}^{8}\\&=0\end{aligned}}$

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