Science:Math Exam Resources/Courses/MATH103/April 2016/Question 09 (c)

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MATH103 April 2016

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Question 09 (c)
Let $x$ be a continuous random variable taking values in $[0,100]$ with probability density function $p(x)$, mean value $\mu$, variance $\sigma ^{2}$, median value $x_{\text{med}}$, and cumulative function $F(x)$. If $F(1/2)={\frac {1}{8}}$, $F(1)={\frac {1}{2}}$, $F(2)={\frac {3}{4}}$, and $F(3)={\frac {7}{8}}$, find the probability that $x$ is greater than $2x_{\text{med}}$.

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
Recall that by the definition of the median, we have $\int _{-\infty }^{x_{\text{med}}}p(x)\,dx={\frac {1}{2}}.$ Relate this definition to the definition of the cumulative function $F(x)$ and the given values.

Hint 2
Recall that the probability that the random variable $x$ takes values in $[a,b]$ is ${\text{P}}[a\leq x\leq b]=\int _{a}^{b}p(x)\,dx.$

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

The question is to find ${\text{P}}[x>2x_{\text{med}}]=\int _{2x_{\text{med}}}^{\infty }p(x)\,dx=\int _{2x_{\text{med}}}^{100}p(x)\,dx.$ We know that $F(x)=\int _{-\infty }^{x}p(t)\,dt,$ and for the random variable in this question $x\in [0,100],$ so $F(x)=\int _{0}^{x}p(t)\,dt.$ We now try to write the desired probability in terms of $F(x):$ ${\begin{aligned}{\text{P}}[x>2x_{\text{med}}]&=\int _{2x_{\text{med}}}^{100}p(x)\,dx\\&=\int _{0}^{100}p(x)dx-\int _{0}^{2x_{\text{med}}}p(x)\,dx\\&=1-F(2x_{\text{med}}).\end{aligned}}$ But ${\color {OrangeRed}F(x_{\text{med}})}=\int _{0}^{x_{\text{med}}}p(x)\,dx={\color {OrangeRed}{\frac {1}{2}}}$ by definition of $x_{\text{med}}.$ Checking the given values in the question, we see that ${\color {OrangeRed}F(1)={\frac {1}{2}}},$ so $x_{\text{med}}=1.$ Therefore ${\text{P}}[x>2x_{\text{med}}]=1-F(2x_{\text{med}})=1-F(2)=1-{\frac {3}{4}}=\color {blue}{\frac {1}{4}}.$

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